Find the following indefinite integral. ∫ − 300 d x \int-300dx ∫ − 300 d x

− 300 x + C -300x+C − 300 x + C

Find the following indefinite integral. ∫ 4 x 3 d x \int4x^3dx ∫ 4 x 3 d x

4 3 x 4 + C \frac43x^4+C 3 4 x 4 + C

Find the following indefinite integral. ∫ 100 x 99 d x \int100x^{99}dx

100 99 x 100 \frac{100}{99}x^{100}

100 99 x 100 + C \frac{100}{99}x^{100}+C

Find h ( x ) h\left(x\right) h ( x ) by evaluating the following indefinite integral. h ( x ) = x 100 d x h\left(x\right)=x^{100}dx h ( x ) = x 100 d x

h ( x ) = x 101 101 + C h\left(x\right)=\frac{x^{101}}{101}+C h ( x ) = 101 x 101 + C

h ( x ) = x 101 100 + C h\left(x\right)=\frac{x^{101}}{100}+C h ( x ) = 100 x 101 + C

h ( x ) = x 101 101 h\left(x\right)=\frac{x^{101}}{101} h ( x ) = 101 x 101

h ( x ) = x 101 100 h\left(x\right)=\frac{x^{101}}{100} h ( x ) = 100 x 101

Find f ( x ) f\left(x\right) f ( x ) by evaluating the following indefinite integral. f ( x ) = ∫ ( 100 x 2 − 35 x − 13 2 ) d x f\left(x\right)=\int\left(100x^2-35x-\frac{13}{2}\right)dx f ( x ) = ∫ ( 100 x 2 − 35 x − 2 13 ) d x

f ( x ) = 100 3 x 3 − 35 2 x 2 − 13 2 x + C f\left(x\right)=\frac{100}{3}x^3-\frac{35}{2}x^2-\frac{13}{2}x+C f ( x ) = 3 100 x 3 − 2 35 x 2 − 2 13 x + C

f ( x ) = 50 x 3 − 35 x 2 − 13 2 x + C f\left(x\right)=50x^3-35x^2-\frac{13}{2}x+C f ( x ) = 50 x 3 − 35 x 2 − 2 13 x + C

f ( x ) = 50 x 3 − 35 x 2 + C f\left(x\right)=50x^3-35x^2+C f ( x ) = 50 x 3 − 35 x 2 + C

f ( x ) = 100 x 3 − 35 x 2 − 13 2 x + C f\left(x\right)=100x^3-35x^2-\frac{13}{2}x+C f ( x ) = 100 x 3 − 35 x 2 − 2 13 x + C

Find f ( x ) f\left(x\right) f ( x ) by evaluating the following indefinite integral. f ( x ) = ∫ ( 8 x 7 + 10 x − 20 ) d x f\left(x\right)=\int\left(8x^7+10x-20\right)dx

f ( x ) = x 8 + 5 x 2 − 20 x + C f\left(x\right)=x^8+5x^2-20x+C

f ( x ) = 8 x 8 + 10 x 2 − 20 x + C f\left(x\right)=8x^8+10x^2-20x+C

f ( x ) = x 8 + 5 x 2 + C f\left(x\right)=x^8+5x^2+C

f ( x ) = 8 x 8 + 10 x 2 + C f\left(x\right)=8x^8+10x^2+C

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals

7. Antiderivatives & Indefinite Integrals

Indefinite Integrals: Videos & Practice Problems

Video Lessons Practice Worksheet

Topic summary

Indefinite integrals represent the reverse process of derivatives, utilizing the same principles as antiderivatives. The integral of a function, denoted by an elongated 's' symbol, involves finding the antiderivative and adding a constant of integration, c. Key rules include the power rule, where the exponent is increased by one and divided by the new exponent, and the sum and difference rule, allowing separate integration of combined functions. Understanding these concepts is essential for solving more complex integrals, particularly polynomials, efficiently.

concept

Introduction to Indefinite Integrals

Video duration:

Introduction to Indefinite Integrals Video Summary

In calculus, the concept of the indefinite integral is closely related to antiderivatives, serving as the reverse process of differentiation. The notation for an indefinite integral is represented by an elongated 'S' symbol, indicating the operation of finding the antiderivative of a function, known as the integrand. When integrating a function, denoted as f(x), the result is the antiderivative F(x) plus a constant c, referred to as the constant of integration. This constant arises because the derivative of any constant is zero, meaning multiple functions can yield the same derivative.

The variable of integration, indicated by dx, specifies the variable with respect to which the integration is performed. For example, if integrating with respect to x, the integrand might be a function like 3x². The process of finding the indefinite integral mirrors that of finding the antiderivative, with the addition of this new notation.

To illustrate the concept, consider the following examples:

1. **Indefinite Integral of 0**: The integral of 0 is simply a constant, c, since the antiderivative of a constant is zero.

2. **Indefinite Integral of a Constant**: For the integral of a constant, such as 3, the result is 3x + c. This is because the antiderivative of a constant a is ax, where x is the variable of integration.

3. **Indefinite Integral of a Polynomial**: When integrating a polynomial like 3x², we apply the power rule in reverse. The antiderivative is x³ + c, as the derivative of x³ yields 3x².

To verify the correctness of these integrals, one can differentiate the resulting functions. For instance, differentiating c gives 0, differentiating 3x + c yields 3, and differentiating x³ + c results in 3x², confirming that the integrals were computed correctly.

Understanding the process of indefinite integration is essential, as it lays the groundwork for more complex calculus concepts and applications.

Problem

Find the following indefinite integral.
$\int-300dx$

$-300x+C$

$-300x$

$300x+C$

$-300+C$

Problem

Find the following indefinite integral.
$\int4x^3dx$

$x^4$

$x^4+C$

$\frac43x^4+C$

$\frac43x^4$

Problem

Find the following indefinite integral.
$\int 100 x^{99} d x$

$\frac{100}{99} x^{100}$

$x to the 100th power$

$\frac{100}{99} x^{100} + C$

$x^{100} + C$

concept

Power Rule for Indefinite Integrals

Video duration:

Power Rule for Indefinite Integrals Video Summary

Indefinite integrals are a fundamental concept in calculus, representing the reverse process of differentiation. Understanding how to find these integrals can be simplified by applying the power rule, which is analogous to the power rule used for derivatives. The power rule for derivatives states that for a function $ f(x) = x^n $, the derivative is given by:

\[f'(x) = n \cdot x^{n-1}\]

In contrast, the power rule for integrals allows us to find the indefinite integral of a function of the form $ x^n $. To apply this rule, we increase the exponent by 1 and then divide by the new exponent. The integral can be expressed as:

\[\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\]

Here, $ C $ represents the constant of integration, which is essential when calculating indefinite integrals. It is important to note that this rule is valid only when $ n \neq -1 $ because division by zero is undefined.

For example, to find the integral of $ x^6 $, we apply the power rule:

\[\int x^6 \, dx = \frac{x^{6+1}}{6+1} + C = \frac{x^7}{7} + C\]

Similarly, for the function $ g(t) = t^4 $, the integral is calculated as follows:

\[\int t^4 \, dt = \frac{t^{4+1}}{4+1} + C = \frac{t^5}{5} + C\]

To verify the correctness of these integrals, one can differentiate the resulting expressions. For instance, differentiating $ \frac{x^7}{7} + C $ will yield the original function $ x^6 $, confirming that the integration process was executed correctly.

In summary, mastering the power rule for integrals provides a straightforward method for solving a wide range of problems involving polynomial functions. This foundational skill is crucial for further studies in calculus and its applications.

Problem

Find $h (x)$ by evaluating the following indefinite integral.
$h (x) = \int x^{8} d x$

$h (x) = x^{9} + C$

$h (x) = \frac{x^{9}}{9} + C$

$h (x) = \frac{x^{9}}{9}$

$h (x) = x^{9}$

Problem

Find $h\left(x\right)$ by evaluating the following indefinite integral.
$h\left(x\right)=x^{100}dx$

$h\left(x\right)=\frac{x^{101}}{101}+C$

$h\left(x\right)=\frac{x^{101}}{100}+C$

$h\left(x\right)=\frac{x^{101}}{101}$

$h\left(x\right)=\frac{x^{101}}{100}$

Problem

Find $h\left(x\right)$ by evaluating the following indefinite integral.
$h (x) = 25 x^{4} d x$

$h (x) = 25 x^{5}$

$h (x) = 5 x^{5}$

$h (x) = 5 x^{5} + C$

$h (x) = 25 x^{5} + C$

concept

Additional Rules for Indefinite Integrals

Video duration:

Additional Rules for Indefinite Integrals Video Summary

In the study of indefinite integrals, understanding how to integrate various functions is crucial. When faced with a combination of functions, such as polynomials, it’s essential to apply specific rules that simplify the integration process. The sum and difference rules, which you may recall from derivatives, are equally applicable to integrals. This means that if you have two functions being added or subtracted, you can integrate them separately. For example, the integral of $x + 6$ can be expressed as the sum of the integrals of $x$ and $6$:

$$\int (x + 6) \, dx = \int x \, dx + \int 6 \, dx$$

Using the power rule for integration, we find:

$$\int x \, dx = \frac{x^2}{2} \quad \text{and} \quad \int 6 \, dx = 6x$$

Thus, the complete integral becomes:

$$\frac{x^2}{2} + 6x + C$$

where $C$ is the constant of integration. It’s important to note that we only need one constant $C$ at the end, as any constants from individual integrals will combine into a single constant.

Another key rule is the constant multiple rule, which allows you to factor out constants from the integral. For instance, if you have the integral of $5x^3$, you can rewrite it as:

$$\int 5x^3 \, dx = 5 \int x^3 \, dx$$

Applying the power rule again gives:

$$\int x^3 \, dx = \frac{x^4}{4}$$

Thus, the integral becomes:

$$5 \cdot \frac{x^4}{4} + C = \frac{5x^4}{4} + C$$

To integrate more complex functions, such as $x^2 - 3x + 6$, you can apply both the sum and difference rule and the constant multiple rule. The integral can be expressed as:

$$\int (x^2 - 3x + 6) \, dx = \int x^2 \, dx - \int 3x \, dx + \int 6 \, dx$$

Breaking it down, we have:

$$\int x^2 \, dx = \frac{x^3}{3}, \quad \int 3x \, dx = 3 \cdot \frac{x^2}{2} = \frac{3x^2}{2}, \quad \text{and} \quad \int 6 \, dx = 6x$$

Combining these results, the integral of the entire function is:

$$\frac{x^3}{3} - \frac{3x^2}{2} + 6x + C$$

This process illustrates how to effectively integrate polynomials by applying the sum and difference rule along with the constant multiple rule. Mastering these techniques will enable you to tackle more complicated functions with confidence.

Problem

Find $f\left(x\right)$ by evaluating the following indefinite integral.
$f\left(x\right)=\int\left(100x^2-35x-\frac{13}{2}\right)dx$

$f\left(x\right)=\frac{100}{3}x^3-\frac{35}{2}x^2-\frac{13}{2}x+C$

$f\left(x\right)=50x^3-35x^2-\frac{13}{2}x+C$

$f\left(x\right)=50x^3-35x^2+C$

$f\left(x\right)=100x^3-35x^2-\frac{13}{2}x+C$

Problem

Find $f\left(x\right)$ by evaluating the following indefinite integral.
$f (x) = \int (8 x^{7} + 10 x - 20) d x$

$f (x) = x^{8} + 5 x^{2} - 20 x + C$

$f (x) = 8 x^{8} + 10 x^{2} - 20 x + C$

$f (x) = x^{8} + 5 x^{2} + C$

$f (x) = 8 x^{8} + 10 x^{2} + C$

Do you want more practice?

We have more practice problems on Indefinite Integrals

Go over this topic definitions with flashcards

More sets

Indefinite Integrals quiz #1
7. Antiderivatives & Indefinite Integrals
13 Terms

Here’s what students ask on this topic:

What is an indefinite integral?

An indefinite integral represents the reverse process of taking a derivative. It involves finding the antiderivative of a function and adding a constant of integration, denoted as 'c'. The notation for an indefinite integral includes an elongated 's' symbol, known as the integral sign, followed by the integrand (the function being integrated) and the differential (e.g., dx). The result of an indefinite integral is a family of functions that differ by a constant. For example, the indefinite integral of f(x) = 3x² is F(x) = x³ + c.

How do you find the indefinite integral of a polynomial function?

To find the indefinite integral of a polynomial function, you apply the power rule for integration. For each term in the polynomial, increase the exponent by 1 and then divide by the new exponent. Don't forget to add the constant of integration, c. For example, to integrate f(x) = 3x² + 2x + 1, you would find the integral of each term separately: ∫3x² dx = x³, ∫2x dx = x², and ∫1 dx = x. Combining these, the indefinite integral is F(x) = x³ + x² + x + c.

What is the power rule for indefinite integrals?

The power rule for indefinite integrals is a method used to integrate functions of the form xⁿ. To apply the power rule, increase the exponent by 1 and then divide by the new exponent. The formula is: ∫xⁿ dx = (xⁿ⁺¹)/(n+1) + c, where n ≠ -1. For example, to integrate x⁴, you would get ∫x⁴ dx = (x⁵)/5 + c.

What is the constant of integration in indefinite integrals?

The constant of integration, denoted as 'c', is an arbitrary constant added to the result of an indefinite integral. It accounts for the fact that the antiderivative of a function is not unique; any constant added to an antiderivative will still yield the same derivative. For example, the indefinite integral of f(x) = 3x² is F(x) = x³ + c, where c can be any real number.

How do you use the sum and difference rule for indefinite integrals?

The sum and difference rule for indefinite integrals allows you to integrate functions that are sums or differences of other functions separately. If you have ∫(f(x) ± g(x)) dx, you can split it into ∫f(x) dx ± ∫g(x) dx. For example, to integrate ∫(x² + 3x) dx, you can write it as ∫x² dx + ∫3x dx. Using the power rule, you get (x³/3) + (3x²/2) + c.

Your Calculus tutors

Indefinite Integrals: Videos & Practice Problems

Introduction to Indefinite Integrals

Introduction to Indefinite Integrals Video Summary

Find the following indefinite integral.∫−300dx\int-300dx∫−300dx

Find the following indefinite integral.∫4x3dx\int4x^3dx∫4x3dx

Find the following indefinite integral.∫100x99dx\int100x^{99}dx

Power Rule for Indefinite Integrals

Power Rule for Indefinite Integrals Video Summary

Find h(x)h\left(x\right) by evaluating the following indefinite integral.h(x)=∫x8dxh\left(x\right)=\int x^8dx

Find h(x)h\left(x\right)h(x) by evaluating the following indefinite integral.h(x)=x100dxh\left(x\right)=x^{100}dxh(x)=x100dx

Find h(x)h\left(x\right)h(x) by evaluating the following indefinite integral.h(x)=25x4dxh\left(x\right)=25x^4dx

Additional Rules for Indefinite Integrals

Additional Rules for Indefinite Integrals Video Summary

Find f(x)f\left(x\right)f(x) by evaluating the following indefinite integral.f(x)=∫(100x2−35x−132)dxf\left(x\right)=\int\left(100x^2-35x-\frac{13}{2}\right)dxf(x)=∫(100x2−35x−213)dx

Find f(x)f\left(x\right)f(x) by evaluating the following indefinite integral.f(x)=∫(8x7+10x−20)dxf\left(x\right)=\int\left(8x^7+10x-20\right)dx

Do you want more practice?

Go over this topic definitions with flashcards

Here’s what students ask on this topic:

What is an indefinite integral?

How do you find the indefinite integral of a polynomial function?

What is the power rule for indefinite integrals?

What is the constant of integration in indefinite integrals?

How do you use the sum and difference rule for indefinite integrals?

Your Calculus tutors

Find the following indefinite integral.
$\int-300dx$

Find the following indefinite integral.
$\int4x^3dx$

Find the following indefinite integral.
$\int 100 x^{99} d x$

Find $h (x)$ by evaluating the following indefinite integral.
$h (x) = \int x^{8} d x$

Find $h\left(x\right)$ by evaluating the following indefinite integral.
$h\left(x\right)=x^{100}dx$

Find $h\left(x\right)$ by evaluating the following indefinite integral.
$h (x) = 25 x^{4} d x$

Find $f\left(x\right)$ by evaluating the following indefinite integral.
$f\left(x\right)=\int\left(100x^2-35x-\frac{13}{2}\right)dx$

Find $f\left(x\right)$ by evaluating the following indefinite integral.
$f (x) = \int (8 x^{7} + 10 x - 20) d x$