Evaluate the indefinite integral. ∫ − ( 6 ) x d x \int_{}^{}-\left(6\right)^{x}dx

− ( 6 ) x ln ⁡ 6 + C -\frac{\left(6\right)^{x}}{\ln6}+C

− x 6 x − 1 + C -x6^{x-1}+C

− 1 x + 1 6 x + 1 + C -\frac{1}{x+1}6^{x+1}+C

− ( 6 ) x + 1 ln ⁡ x + C -\frac{\left(6\right)^{x+1}}{\ln x}+C

Evaluate the indefinite integral. ∫ 3 x 4 − ( 5 ) x d x \int_{}^{}3x^4-\left(5\right)^{x}dx

3 5 x 5 − 5 x ln ⁡ 5 + C \frac35x^5-\frac{5^{x}}{\ln5}+C

3 5 x 5 − 5 x + 1 ln ⁡ 5 + C \frac35x^5-\frac{5^{x+1}}{\ln5}+C

12 x 5 − 5 x ln ⁡ 5 + C 12x^5-\frac{5^{x}}{\ln5}+C

12 x 5 − 5 x + 1 ln ⁡ 5 + C 12x^5-\frac{5^{x+1}}{\ln5}+C

Evaluate the indefinite integral. ∫ 2 ( 1 3 ) x d x \int_{}^{}2\left(\frac13\right)^{x}dx

− 2 3 x ln ⁡ 3 + C -\frac{2}{3^{x}\ln3}+C

− 2 x x + 1 4 + C -\frac{2x^{x+1}}{4}+C

− 2 x ln ⁡ 3 + C -\frac{2x}{\ln3}+C

− 2 x x + 1 3 x ln ⁡ 4 + C -\frac{2x^{x+1}}{3^{x}\ln4}+C

Evaluate the indefinite integral. ∫ ( x − e x ) d x \int_{}^{}\left(\sqrt{x}-e^{x}\right)dx

2 3 x 3 2 − e x + C \frac23x^{\frac32}-e^{x}+C

2 3 x 2 3 − e x + C \frac23x^{\frac23}-e^{x}+C

2 3 x 3 2 − x e x + C \frac23x^{\frac32}-xe^{x}+C

2 3 x 2 3 − 1 x e x + C \frac23x^{\frac23}-\frac{1}{x}e^{x}+C

Evaluate the indefinite integral. ∫ ( 4 e x + 1 x 3 ) d x \int_{}^{}\left(4e^{x}+\frac{1}{x^3}\right)dx

4 e x − 1 2 x 2 + C 4e^{x}-\frac{1}{2x^2}+C

4 e x − 1 4 x 4 + C 4e^{x}-\frac{1}{4x^4}+C

4 x e x − 1 2 x 2 + C 4xe^{x}-\frac{1}{2x^2}+C

4 x e x − 1 4 x 4 + C 4xe^{x}-\frac{1}{4x^4}+C

Evaluate the indefinite integral. ∫ e 2 x x d x \int_{}^{}\frac{e^{2\sqrt{x}}}{\sqrt{x}}dx

5 e 2 x 3 x 3 2 + C \frac{5e^{2\sqrt{x}}}{3x^{\frac32}}+C

3 e x 7 2 7 x 3 2 + C \frac{3e^{x^{\frac72}}}{7x^{\frac32}}+C

7 e x 7 2 3 x 3 2 + C \frac{7e^{x^{\frac72}}}{3x^{\frac32}}+C

Evaluate the indefinite integral. ∫ e sec ⁡ θ sec ⁡ θ tan ⁡ θ d θ \int_{}^{}e^{\sec\theta}\sec\theta\tan\theta d\theta

e sec ⁡ θ + C e^{\sec\theta}+C

− e sec ⁡ θ c o s 2 ⁡ θ + C -\frac{e^{\sec\theta}}{cos^2⁡\theta}+C

− e sec ⁡ θ + C -e^{\sec\theta}+C

e sec ⁡ θ c o s 2 ⁡ θ + C \frac{e^{\sec\theta}}{cos^2⁡\theta}+C

11. Integrals of Inverse, Exponential, & Logarithmic Functions

Integrals of Exponential Functions

11. Integrals of Inverse, Exponential, & Logarithmic Functions

Integrals of Exponential Functions: Videos & Practice Problems

Learn Concepts Practice Worksheet

Topic summary

```html

To find the integral of an exponential function $b^{x}$ , divide by the natural log of the base: $\frac{b^{x}}{\ln (b)}$ plus a constant. For $e^{x}$ , the integral is $e^{x}$ itself. Use rules like the constant multiple rule and sum rule to simplify integrals. This approach is crucial for understanding integrals in calculus.

```

concept

Integrals of General Exponential Functions

Video duration:

Integrals of General Exponential Functions Video Summary

In calculus, understanding the relationship between derivatives and integrals is crucial, especially when dealing with exponential functions. The integral of a general exponential function, expressed as $ b^x $, can be derived by reversing the process used to find its derivative. When differentiating $ b^x $, the result is $ b^x \cdot \ln(b) $. Therefore, to find the integral of $ b^x $, we divide $ b^x $ by $ \ln(b) $. The formula for the indefinite integral is given by:

\[\int b^x \, dx = \frac{b^x}{\ln(b)} + C\]

Here, $ C $ represents the constant of integration, and it is important to note that the base $ b $ must be greater than zero and not equal to one, as these conditions define a valid exponential function.

To illustrate this rule, consider the integral of $ 7^x $. Applying the formula, we find:

\[\int 7^x \, dx = \frac{7^x}{\ln(7)} + C\]

Next, we can apply this rule to more complex integrals involving sums of exponential functions. For example, to evaluate the integral of $ 3 \cdot \left(\frac{1}{2}\right)^x + 8^x $, we can separate the integral into two parts using the sum rule:

\[\int \left(3 \cdot \left(\frac{1}{2}\right)^x + 8^x\right) \, dx = \int 3 \cdot \left(\frac{1}{2}\right)^x \, dx + \int 8^x \, dx\]

Utilizing the constant multiple rule, we can factor out the constant 3 from the first integral:

\[= 3 \int \left(\frac{1}{2}\right)^x \, dx + \int 8^x \, dx\]

Now, applying the integral rule for each term, we have:

\[= 3 \cdot \frac{\left(\frac{1}{2}\right)^x}{\ln\left(\frac{1}{2}\right)} + \frac{8^x}{\ln(8)} + C\]

Next, we can simplify the expression further using properties of logarithms and exponents. The term $ \left(\frac{1}{2}\right)^x $ can be rewritten as $ \frac{1^x}{2^x} $, and the natural logarithm of a fraction can be expressed as:

\[\ln\left(\frac{1}{2}\right) = \ln(1) - \ln(2) = -\ln(2)\]

Thus, the integral simplifies to:

\[= -\frac{3}{2^x \cdot \ln(2)} + \frac{8^x}{\ln(8)} + C\]

In conclusion, mastering the integration of exponential functions allows for the application of various rules and properties, enhancing problem-solving skills in calculus. As you continue practicing these concepts, you'll gain confidence in handling more complex integrals involving exponential functions.

Problem

Evaluate the indefinite integral.
$\int_{}^{} - {(6)}^{x} d x$

$- x 6^{x - 1} + C$

$- \frac{1}{x + 1} 6^{x + 1} + C$

$- \frac{{(6)}^{x}}{\ln 6} + C$

$- \frac{{(6)}^{x + 1}}{\ln x} + C$

Problem

Evaluate the indefinite integral.
$\int_{}^{} 3 x^{4} - {(5)}^{x} d x$

$\frac{3}{5} x^{5} - \frac{5^{x + 1}}{\ln 5} + C$

$\frac{3}{5} x^{5} - \frac{5^{x}}{\ln 5} + C$

$12 x^{5} - \frac{5^{x}}{\ln 5} + C$

$12 x^{5} - \frac{5^{x + 1}}{\ln 5} + C$

Problem

Evaluate the indefinite integral.
$\int_{}^{} 2 {(\frac{1}{3})}^{x} d x$

$- \frac{2}{3^{x} \ln 3} + C$

$- \frac{2 x^{x + 1}}{4} + C$

$- \frac{2 x}{\ln 3} + C$

$- \frac{2 x^{x + 1}}{3^{x} \ln 4} + C$

example

Integrals of General Exponential Functions Example 1

Video duration:

Integrals of General Exponential Functions Example 1 Video Summary

To evaluate integrals involving exponential functions with a variable in the exponent, substitution is often necessary. For instance, consider the integral of $4^{7x} \, dx$. Here, the exponent is a function of $x$, specifically $7x$. To simplify the integration process, we set $u = 7x$. Consequently, the differential $du$ becomes $7 \, dx$, or $dx = \frac{du}{7}$. To maintain the integrity of the integral while substituting, we multiply the integral by $7$ and its reciprocal, $\frac{1}{7}$, leading to the expression:\[\int 4^{7x} \, dx = \frac{1}{7} \int 4^u \, du\]Applying the rule for integrating exponential functions, we find:\[\frac{1}{7} \cdot \frac{4^u}{\ln(4)} + C\]Substituting back for $u$, we arrive at the final result:\[\frac{4^{7x}}{7 \ln(4)} + C\]This method of substitution is applicable to any general exponential function where the exponent is a function of another variable, yielding the formula $\frac{b^u}{\ln(b)}$ for integration.

In another example, we evaluate the integral of $5^{\cos(\theta)} \sin(\theta) \, d\theta$. Here, we set $u = \cos(\theta)$, which gives us $du = -\sin(\theta) \, d\theta$. This allows us to rewrite the integral as:\[-\int 5^u \, du\]Using the integration rule for exponential functions, we find:\[-\frac{5^u}{\ln(5)} + C\]Substituting back for $u$, the final answer is:\[-\frac{5^{\cos(\theta)}}{\ln(5)} + C\]Through these examples, we see that using substitution simplifies the integration of exponential functions, allowing us to apply the general rules effectively. This technique is essential for handling integrals that initially appear complex, particularly when involving trigonometric functions or composite exponents.

Problem

Evaluate the definite integral.
$\int_{0}^{5} \frac{3^{\sqrt{x + 4}}}{\sqrt{x + 4}} d x$

49.15

73.73

77.91

32.77

Problem

Find the area under the graph of $f (x) = 3^{2 x}$ between $x equals 0$ and $x equals 2$ .

36.41

12.29

36.86

12.98

concept

Integrals of Natural Exponential Functions (e^x)

Video duration:

Integrals of Natural Exponential Functions (e^x) Video Summary

In calculus, integrating the exponential function $ e^x $ is a fundamental concept. The integral of $ e^x $ can be derived from the general rule for exponential functions, which states that the integral of $ b^x $ is given by:

\[\int b^x \, dx = \frac{b^x}{\ln(b)} + C\]

For the specific case where the base $ b $ is the mathematical constant $ e $, the integral simplifies significantly. Since $ \ln(e) = 1 $, the integral of $ e^x $ becomes:

\[\int e^x \, dx = e^x + C\]

This result is particularly interesting because it shows that the integral of $ e^x $ is the same as the function itself, reinforcing the unique properties of the exponential function.

When dealing with integrals that include constants multiplied by $ e^x $, such as $ 5e^x $, the constant multiple rule allows us to factor out the constant. Thus, we have:

\[\int 5e^x \, dx = 5 \int e^x \, dx = 5e^x + C\]

To illustrate the application of these rules, consider the integral of a more complex function, such as $ 3x^4 - \pi e^x + 2 $. This can be broken down into separate integrals using the sum and difference rules:

\[\int (3x^4 - \pi e^x + 2) \, dx = \int 3x^4 \, dx - \int \pi e^x \, dx + \int 2 \, dx\]

Now, we can evaluate each integral individually:

1. For $ \int 3x^4 \, dx $, applying the constant multiple rule and the power rule gives:

\[\int 3x^4 \, dx = 3 \cdot \frac{x^5}{5} = \frac{3}{5}x^5\]

2. For $ \int \pi e^x \, dx $, we factor out $ \pi $ and use the integral of $ e^x $:

\[\int \pi e^x \, dx = \pi \int e^x \, dx = \pi e^x\]

3. For $ \int 2 \, dx $, we simply get:

\[\int 2 \, dx = 2x\]

Combining these results, we arrive at the final answer for the integral:

\[\int (3x^4 - \pi e^x + 2) \, dx = \frac{3}{5}x^5 - \pi e^x + 2x + C\]

This example demonstrates the application of various integration rules, including the constant multiple rule, the power rule, and the integral of the exponential function, allowing for the evaluation of more complex integrals effectively.

Problem

Evaluate the indefinite integral.
$\int_{}^{} (\sqrt{x} - e^{x}) d x$

$\frac{2}{3} x^{\frac{2}{3}} - e^{x} + C$

$\frac{2}{3} x^{\frac{3}{2}} - x e^{x} + C$

$\frac{2}{3} x^{\frac{3}{2}} - e^{x} + C$

$\frac{2}{3} x^{\frac{2}{3}} - \frac{1}{x} e^{x} + C$

Problem

Evaluate the indefinite integral.
$\int_{}^{} (4 e^{x} + \frac{1}{x^{3}}) d x$

$4 e^{x} - \frac{1}{4 x^{4}} + C$

$4 x e^{x} - \frac{1}{2 x^{2}} + C$

$4 x e^{x} - \frac{1}{4 x^{4}} + C$

$4 e^{x} - \frac{1}{2 x^{2}} + C$

example

Integrals of Natural Exponential Functions (e^x) Example 2

Video duration:

Integrals of Natural Exponential Functions (e^x) Example 2 Video Summary

To evaluate the indefinite integral of the function $3x^2 e^{x^3} \, dx$, we can utilize the method of substitution, which simplifies the process significantly. When dealing with an exponential function that contains a composite function in its exponent, it is often beneficial to set that composite function as our substitution variable.

In this case, we can let $u = x^3$. Consequently, the differential $du$ is derived from the derivative of $u$, which gives us:

\[du = 3x^2 \, dx\]

This means that $3x^2 \, dx$ can be replaced with $du$ in our integral. Thus, we can rewrite the original integral in terms of $u$ and $du$:

\[\int 3x^2 e^{x^3} \, dx = \int e^u \, du\]

We know from calculus that the integral of $e^u$ is simply $e^u$ plus a constant of integration $C$. Therefore, we have:

\[\int e^u \, du = e^u + C\]

Substituting back for $u$ gives us the final result:

\[e^{x^3} + C\]

This process illustrates how substitution can transform a seemingly complex integral into a straightforward calculation. The key takeaway is that when faced with an integral involving an exponential function, identifying the appropriate substitution can greatly simplify the evaluation.

Problem

Evaluate the indefinite integral.
$\int_{}^{} \frac{e^{2 \sqrt{x}}}{\sqrt{x}} d x$

$\frac{5 e^{2 \sqrt{x}}}{3 x^{\frac{3}{2}}} + C$

$\frac{3 e^{x^{\frac{7}{2}}}}{7 x^{\frac{3}{2}}} + C$

$e^{2 \sqrt{x}} + C$

$\frac{7 e^{x^{\frac{7}{2}}}}{3 x^{\frac{3}{2}}} + C$

Problem

Evaluate the indefinite integral.
$\int_{}^{} e^{\sec θ} \sec θ \tan θ d θ$

$- \frac{e^{\sec θ}}{c o s^{2} θ} + C$

$e^{\sec θ} + C$

$- e^{\sec θ} + C$

$\frac{e^{\sec θ}}{c o s^{2} θ} + C$

example

Integrals of Natural Exponential Functions (e^x) Example 3

Video duration:

Integrals of Natural Exponential Functions (e^x) Example 3 Video Summary

To evaluate integrals using substitution, we often start by identifying functions within the integrand that can simplify the process. For the first integral, we have:

$$\int \frac{e^t + e^{4t}}{e^{2t}} \, dt$$

Recognizing that we can split the integrand into two separate fractions allows us to simplify further:

$$\int \left( \frac{e^t}{e^{2t}} + \frac{e^{4t}}{e^{2t}} \right) dt = \int \left( e^{t - 2t} + e^{4t - 2t} \right) dt = \int \left( e^{-t} + e^{2t} \right) dt$$

This leads us to two separate integrals:

$$\int e^{-t} \, dt + \int e^{2t} \, dt$$

For the first integral, we can use the substitution $ u = -t $, which gives $ du = -dt $. This transforms the integral into:

$$-\int e^u \, du = -e^u + C = -e^{-t} + C$$

For the second integral, we set $ u = 2t $, leading to $ du = 2dt $ or $ dt = \frac{du}{2} $. Thus, we rewrite the integral as:

$$\frac{1}{2} \int e^u \, du = \frac{1}{2} e^u + C = \frac{1}{2} e^{2t} + C$$

Combining these results, the final answer for the first integral is:

$$-e^{-t} + \frac{1}{2} e^{2t} + C$$

Next, we evaluate the second integral:

$$\int 3e^{3x}(1 + e^{3x})^5 \, dx$$

Here, a strategic substitution is key. Instead of letting $ u = 3x $, we choose $ u = 1 + e^{3x} $. The derivative of this function is:

$$du = 3e^{3x} \, dx$$

This matches the remaining part of our integral, allowing us to rewrite it as:

$$\int u^5 \, du$$

Applying the power rule for integration gives us:

$$\frac{1}{6} u^6 + C$$

Substituting back for $ u $, we find:

$$\frac{1}{6} (1 + e^{3x})^6 + C$$

In summary, while the instinct may be to choose the exponent as the substitution, sometimes a more complex function within the integrand provides a clearer path to simplification. Mastering these techniques enhances our ability to tackle a variety of integral problems effectively.

Problem

Find the definite integral.
$\int_{- 1}^{1} x^{2} e^{x^{3}} d x$

0.78

0.86

0.31

-0.74

Problem

Find the area of $f (x) = 3 + e^{2 x}$ from $x equals 0$ to $x equals 2$ .

-20.80

32.80

-21.80

-21.30

Do you want more practice?

We have more practice problems on Integrals of Exponential Functions

Here’s what students ask on this topic:

How do you find the integral of an exponential function b^x?

To find the integral of an exponential function b^x, you reverse the derivative process. The integral is given by the formula:

\frac{b^{x}}{\ln (b)} + c

where b is the base of the exponential function, ln(b) is the natural logarithm of b, and c is the constant of integration. This formula is applicable when b is greater than zero and not equal to one. Understanding this rule allows you to integrate more complex functions involving exponential terms and constants.

What is the integral of e^x?

The integral of e^x is a special case of the general exponential function integral. Since the base is e, the integral simplifies to:

e^{x} + c

where c is the constant of integration. This result makes sense because the derivative of e^x is also e^x. Therefore, when integrating e^x, you end up with the same function plus the constant of integration. This property is useful in solving more complex integrals involving e^x and other terms.

How do you apply the constant multiple rule in integrals?

The constant multiple rule in integrals allows you to factor out constants from the integral. If you have an integral of the form:

\int k f (x) dx

where k is a constant, you can rewrite it as:

k \int f (x) dx

This simplifies the integration process by allowing you to focus on integrating the function f(x) separately. This rule is particularly useful when dealing with integrals involving exponential functions and other terms.

What are the restrictions on the base b for the integral of b^x?

When finding the integral of b^x, there are certain restrictions on the base b. Specifically, b must be greater than zero and not equal to one. These restrictions are based on the definition of a general exponential function. If b were equal to one, the function would be constant, and the integral would not follow the standard exponential rules. Similarly, if b were less than or equal to zero, the function would not be defined in the real number system. These restrictions ensure the function behaves as expected in calculus.

How do you integrate a function involving multiple exponential terms?

To integrate a function involving multiple exponential terms, you can use the sum rule to split the integral into separate parts. For example, if you have:

\int (a b^{x} + c^{x}) dx

you can rewrite it as:

a \int b^{x} dx + \int c^{x} dx

Then, apply the exponential integral rule to each term separately. This approach allows you to handle each exponential term individually, making the integration process more manageable.

Your Calculus tutors

Integrals of Exponential Functions: Videos & Practice Problems

Integrals of General Exponential Functions

Integrals of General Exponential Functions Video Summary

Evaluate the indefinite integral. ∫−(6)xdx\int_{}^{}-\left(6\right)^{x}dx

Evaluate the indefinite integral.∫3x4−(5)xdx\int_{}^{}3x^4-\left(5\right)^{x}dx

Evaluate the indefinite integral.∫2(13)xdx\int_{}^{}2\left(\frac13\right)^{x}dx

Integrals of General Exponential Functions Example 1

Integrals of General Exponential Functions Example 1 Video Summary

Evaluate the definite integral.∫053x+4x+4dx\int_0^5\frac{3^{\sqrt{x+4}}}{\sqrt{x+4}}dx

Find the area under the graph of f(x)=32xf\left(x\right)=3^{2x} between x=0x=0 and x=2x=2.

Integrals of Natural Exponential Functions (e^x)

Integrals of Natural Exponential Functions (e^x) Video Summary

Evaluate the indefinite integral.∫(x−ex)dx\int_{}^{}\left(\sqrt{x}-e^{x}\right)dx

Evaluate the indefinite integral.∫(4ex+1x3)dx\int_{}^{}\left(4e^{x}+\frac{1}{x^3}\right)dx

Integrals of Natural Exponential Functions (e^x) Example 2

Integrals of Natural Exponential Functions (e^x) Example 2 Video Summary

Evaluate the indefinite integral.∫e2xxdx\int_{}^{}\frac{e^{2\sqrt{x}}}{\sqrt{x}}dx

Evaluate the indefinite integral.∫esecθsecθtanθdθ\int_{}^{}e^{\sec\theta}\sec\theta\tan\theta d\theta

Integrals of Natural Exponential Functions (e^x) Example 3

Integrals of Natural Exponential Functions (e^x) Example 3 Video Summary

Find the definite integral.∫−11x2ex3dx\int_{-1}^1x^2e^{x^3}dx

Find the area of f(x)=3+e2xf\left(x\right)=3+e^{2x} from x=0x=0 to x=2x=2.

Do you want more practice?

Here’s what students ask on this topic:

How do you find the integral of an exponential function b^x?

What is the integral of e^x?

How do you apply the constant multiple rule in integrals?

What are the restrictions on the base b for the integral of b^x?

How do you integrate a function involving multiple exponential terms?

Your Calculus tutors

Evaluate the indefinite integral.
$\int_{}^{} - {(6)}^{x} d x$

Evaluate the indefinite integral.
$\int_{}^{} 3 x^{4} - {(5)}^{x} d x$

Evaluate the indefinite integral.
$\int_{}^{} 2 {(\frac{1}{3})}^{x} d x$

Evaluate the definite integral.
$\int_{0}^{5} \frac{3^{\sqrt{x + 4}}}{\sqrt{x + 4}} d x$

Find the area under the graph of $f (x) = 3^{2 x}$ between $x equals 0$ and $x equals 2$ .

Evaluate the indefinite integral.
$\int_{}^{} (\sqrt{x} - e^{x}) d x$

Evaluate the indefinite integral.
$\int_{}^{} (4 e^{x} + \frac{1}{x^{3}}) d x$

Evaluate the indefinite integral.
$\int_{}^{} \frac{e^{2 \sqrt{x}}}{\sqrt{x}} d x$

Evaluate the indefinite integral.
$\int_{}^{} e^{\sec θ} \sec θ \tan θ d θ$

Find the definite integral.
$\int_{- 1}^{1} x^{2} e^{x^{3}} d x$

Find the area of $f (x) = 3 + e^{2 x}$ from $x equals 0$ to $x equals 2$ .