Integration Review: Antiderivatives and Substitution Techniques

7.0 Integration Review

Objectives of Integration Review

A comprehensive review of integration techniques, focusing on antiderivatives, the power rule, standard antiderivatives, and integration by substitution. Both indefinite and definite integrals are considered, with special attention to trigonometric and exponential functions.

• Antiderivatives using Power Rule

• Standard Antiderivatives

• Integrating Functions by Substitution

• Definite Integrals by Substitution

Antiderivatives & the Power Rule

Antiderivative

An antiderivative of a function f(x) is a function F(x) such that F'(x) = f(x). The process of finding an antiderivative is called integration.

Power Rule for Integration

The Power Rule is used to find the antiderivative of functions of the form x^n where n ≠ -1:

• For any real number n ≠ -1:

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

• C is the constant of integration.

Examples

1. Find the antiderivative of f(x) = x :

   $\int x \, dx = \frac{x^2}{2} + C$

2. Find the antiderivative of $f\left(x\right)=x^3$:

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

3. Find the antiderivative of f(x) = 3x^2 + x - 7:

   $\int (3x^2 + x - 7) \, dx = x^3 + \frac{x^2}{2} - 7x + C$

Standard Antiderivatives

Common Functions and Their Antiderivatives

Many functions have well-known antiderivatives. The following table summarizes some of the most common ones:

Function	Particular Antiderivative	Function	Particular Antiderivative
$x^n$	$\frac{x^{n+1}}{n+1}$, $n \neq -1$	$\sec^2 x$	$\tan x$
$e^x$	$e^x$	$\csc^2 x$	$-\cot x$
$\frac{1}{x}$	$\ln|x|$	$\sec x \tan x$	$\sec x$
$\sin x$	$-\cos x$	$\csc x \cot x$	$-\csc x$
$\cos x$	$\sin x$

Examples

1. Find the antiderivative of f(x) = e^x:

   $\int e^x \, dx = e^x + C$

2. Find the antiderivative of f(x) = \sin(x):

   $\int \sin(x) \, dx = -\cos(x) + C$

3. Find the antiderivative of f(x) = \csc^2(x):

   $\int \csc^2(x) \, dx = -\cot(x) + C$

Integration by Substitution

Substitution Method

The substitution method (also called u-substitution) is used to simplify integrals by substituting part of the integrand with a new variable. This is especially useful when the integrand is a composite function.

1. Let $u = g(x)$, then $\frac{du}{dx} = g'(x)$, so $du = g'(x)dx$.

2. Rewrite the integral in terms of $u$ and $du$.

3. Integrate with respect to $u$, then substitute back $x$.

Examples

Find $\int x^2 \sec(x^3) dx$:

1. Let $u = x^3$, so $du = 3x^2 dx$ or $x^2 dx = \frac{du}{3}$. $\int x^2 \sec(x^3) dx = \int \sec(u) \frac{du}{3} = \frac{1}{3} \int \sec(u) du = \frac{1}{3} \ln|\sec(u) + \tan(u)| + C = \frac{1}{3} \ln|\sec(x^3) + \tan(x^3)| + C$

2. Find $\int \frac{3 + x}{1 + x^2} dx$:

   Split the fraction: $\int \frac{3}{1 + x^2} dx + \int \frac{x}{1 + x^2} dx$ $= 3 \arctan(x) + \frac{1}{2} \ln|1 + x^2| + C$

Definite Integrals and the Substitution Rule

Definite Integral

A definite integral computes the net area under a curve between two points a and b:

$\int_a^b f(x) dx$

Substitution Rule for Definite Integrals

If $u = g(x)$ is a differentiable function whose range is an interval $I$, and $f$ is continuous on I, then:

$\int_a^b f(g(x))g'(x) dx = \int_{g(a)}^{g(b)} f(u) du$

Examples

1. Evaluate $\int_0^2 x^2 \sqrt{1 + x^2} dx$:

2. Let $u = 1 + x^2$, $du = 2x dx$, so $x dx = \frac{du}{2}$.

3. Adjust limits: when $x = 0$, $u = 1$; when $x = 2$, $u = 5$. $\int_0^2 x^2 \sqrt{1 + x^2} dx = \int_{1}^{5} (u - 1)^{1/2} \frac{du}{2}$ (after further substitution and simplification).

4. Evaluate $\int_{-2}^2 \frac{1}{(x-4)^2} dx$:

5. Use substitution $u = x - 4$, $du = dx$, limits change accordingly.

Common Antiderivative Quiz Practice

Practice Problems

Given a function, find its antiderivative:

1. $f(x) = \sin(x)$

2. $f(x) = \cos(x)$

3. $f(x) = \tan(x)$

4. $f(x) = \sec^2(x)$

5. $f(x) = \sec(x)\tan(x)$

6. $f(x) = \csc^2(x)$

7. $f(x) = \csc(x)\cot(x)$

8. $f(x) = \sinh(x)$

9. $f(x) = \cosh(x)$

10. $f(x) = \tanh(x)$

11. $f(x) = \text{sech}^2(x)$

12. $f(x) = \tan^{-1}(x)$

Refer to the table above for their antiderivatives.

Summary Table: Common Antiderivatives

Function $f\left(x\right)$	Antiderivative $F\left(x\right)$
$x^n$	$\frac{x^{n+1}}{n+1} + C$
$e^x$	$e^x + C$
$\sin(x)$	$-\cos(x) + C$
$\cos(x)$	$\sin(x) + C$
$\sec^2(x)$	$\tan(x) + C$
$\csc^2(x)$	$-\cot(x) + C$
$\sec(x)\tan(x)$	$\sec(x) + C$
$\csc(x)\cot(x)$	$-\csc(x) + C$

Additional Practice: Integration by Substitution

1. $\int 6x(x^2 + 2)^2 dx$

2. $\int 6xe^{x^2} dx$

3. $\int 3x^2 - 4 dx$

4. $\int \sqrt{2x-1} dx$

5. $\int x^2 \sqrt{x^2 - 1} dx$

6. $\int x\sqrt{2x-1} dx$

7. $\int \sin(3x) dx$

8. $\int \sin^2(3x)\cos(3x) dx$

Apply substitution and trigonometric identities as appropriate.

Key Takeaways

• Use the power rule for polynomials and simple powers of x.

• Memorize standard antiderivatives for exponential, logarithmic, and trigonometric functions.

• Apply substitution for more complex integrals, especially when the integrand is a composite function.

• For definite integrals, always adjust the limits when substituting variables.

Additional info: The notes include both practice problems and summary tables, making them suitable for exam preparation and self-study.