Integration and Antiderivatives

Integration: Finding Antiderivatives

Power Rule for Derivatives

The power rule states that the derivative of $x^n$ is $nx^{n-1}$.

Power Rule for Integration

The power rule for finding antiderivatives is:

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

Where:

• $n$ is a constant exponent.
• $C$ is the constant of integration.

Examples

Example 1: Antiderivative of $3x^2$

1. Add 1 to the exponent: $2 + 1 = 3$. The expression becomes $3x^3$.
2. Divide by the new exponent: $\frac{3x^3}{3} = x^3$.
3. Add the constant of integration: $x^3 + C$.

Example 2: Antiderivative of $x^4$

$\int x^4 dx = \frac{x^{4+1}}{4+1} + C = \frac{x^5}{5} + C = \frac{1}{5}x^5 + C$

Example 3: Antiderivative of $x^2$

$\int x^2 dx = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$

Example 4: Antiderivative of $x^7$

$\int x^7 dx = \frac{x^{7+1}}{7+1} + C = \frac{x^8}{8} + C$

Example 5: Antiderivative of $x$

$\int x dx = \frac{x^{1+1}}{1+1} + C = \frac{x^2}{2} + C$

Antiderivative with Fractions

Example 1: Antiderivative of $\frac{x}{4}$$\int \frac{x}{4} dx = \frac{1}{4} \int x dx = \frac{1}{4} \cdot \frac{x^2}{2} + C = \frac{1}{8}x^2 + C$

Example 2: Antiderivative of $8x^3$

$\int 8x^3 dx = 8 \int x^3 dx = 8 \cdot \frac{x^{3+1}}{3+1} + C = 8 \cdot \frac{x^4}{4} + C = 2x^4 + C$

Antiderivative of a Constant

To find the antiderivative of a constant, add the variable of integration to the constant:

$\int 4 dx = 4x + C$
$\int 5 dy = 5y + C$

Explanation

We can rewrite 4 as $4x^0$, because any number to the power of 0 is 1, so $4 \cdot 1 = 4$.

Then, following the power rule, we have $\frac{4x^{0+1}}{0+1} = \frac{4x^1}{1} = 4x$

Example

$\int -7 dz = -7z + C$

Antiderivative of a Binomial

To find the antiderivative of a binomial, integrate each term separately:

$\int (7x - 6) dx = \int 7x dx - \int 6 dx = \frac{7x^2}{2} - 6x + C$

Antiderivative of a Trinomial

$\int (6x^2 + 4x - 7) dx = \int 6x^2 dx + \int 4x dx - \int 7 dx = \frac{6x^3}{3} + \frac{4x^2}{2} - 7x + C = 2x^3 + 2x^2 - 7x + C$

Antiderivatives of Radical Functions

Example 1: Antiderivative of $\sqrt{x}$

1. Rewrite the radical as a rational exponent: $\sqrt{x} = x^{\frac{1}{2}}$.
2. Apply the power rule:
   $\int x^{\frac{1}{2}} dx = \frac{x^{\frac{1}{2} + 1}}{\frac{1}{2} + 1} + C = \frac{x^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{3} x^{\frac{3}{2}} + C$
3. Rewrite back in radical form: $\frac{2}{3} \sqrt{x^3} + C$

Example 2: Antiderivative of $\sqrt[3]{x^4}$

1. Rewrite the radical as a rational exponent: $\sqrt[3]{x^4} = x^{\frac{4}{3}}$.
2. Apply the power rule:

$\int x^{\frac{4}{3}} dx = \frac{x^{\frac{4}{3} + 1}}{\frac{4}{3} + 1} + C = \frac{x^{\frac{7}{3}}}{\frac{7}{3}} + C = \frac{3}{7} x^{\frac{7}{3}} + C$

3. Rewrite back in radical form: $\frac{3}{7} \sqrt[3]{x^7} + C$

Example 3: Antiderivative of $\sqrt[4]{x^7}$

1. Rewrite the radical as a rational exponent: $\sqrt[4]{x^7} = x^{\frac{7}{4}}$.
2. Apply the power rule:

$\int x^{\frac{7}{4}} dx = \frac{x^{\frac{7}{4} + 1}}{\frac{7}{4} + 1} + C = \frac{x^{\frac{11}{4}}}{\frac{11}{4}} + C = \frac{4}{11} x^{\frac{11}{4}} + C$

3. Rewrite back in radical form: $\frac{4}{11} \sqrt[4]{x^{11}} + C$

Integrals of Trigonometric Expressions

• $\int \cos(x) dx = \sin(x) + C$
• $\int -\sin(x) dx = \cos(x) + C$
• $\int \sin(x) dx = -\cos(x) + C$
• $\int \sec^2(x) dx = \tan(x) + C$
• $\int -\csc^2(x) dx = \cot(x) + C$
• $\int \csc^2(x) dx = -\cot(x) + C$
• $\int \sec(x)\tan(x) dx = \sec(x) + C$
• $\int \csc(x)\cot(x) dx = -\csc(x) + C$

Example

$\int (4 \sin(x) - 5 \cos(x) + 3 \sec^2(x)) dx = -4 \cos(x) - 5 \sin(x) + 3 \tan(x) + C$

Indefinite vs. Definite Integrals

• Indefinite Integral: Results in an algebraic expression with a constant of integration (C). $\int f(x) dx = F(x) + C$
• Definite Integral: Results in a numerical value. It has a lower and upper limit of integration. $\int_a^b f(x) dx = F(b) - F(a)$
  The constant of integration is not needed for definite integrals because it cancels out during evaluation.

Example of Indefinite Integral

$\int 6x^2 dx = 2x^3 + C$

Example of Definite Integral

$\int<em>1^2 6x^2 dx = [2x^3]</em>1^2 = 2(2^3) - 2(1^3) = 16 - 2 = 14$

Example 2

$\int<em>2^3 (8x - 3) dx = [4x^2 - 3x]</em>2^3 = (4(3^2) - 3(3)) - (4(2^2) - 3(2)) = (36 - 9) - (16 - 6) = 27 - 10 = 17$

Fundamental Theorem of Calculus

The fundamental theorem of calculus states that if $f(x)$ is a continuous function on the closed interval $[a, b]$, then:

$\int_a^b f(x) dx = F(b) - F(a)$

Where $F(x)$ is the antiderivative of $f(x)$, i.e., $F'(x) = f(x)$.

Also,

$\int f(x) dx = F(x) + C$

Exponential Functions

The derivative of $e^u$ is $e^u \cdot u'$. For antiderivatives, if $u$ is a linear function $ax + b$, then:

$\int e^u dx = \frac{e^u}{u'} + C$

Where $u'$ is the derivative of $u$ with respect to $x$.

Example 1

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

Example 2

$\int e^{5x} dx = \frac{e^{5x}}{5} + C$

Example 3

$\int e^{-7x} dx = \frac{e^{-7x}}{-7} + C$

Example 4

$\int e^{3x - 5} dx = \frac{e^{3x - 5}}{3} + C$

U-Substitution

U-substitution is a technique used to find the antiderivative of composite functions.

Example 1

Find the antiderivative of $e^{8x}$.

1. Let $u = 8x$.
2. Find $\frac{du}{dx} = 8$, so $du = 8 dx$.
3. Solve for $dx$: $dx = \frac{du}{8}$.
4. Substitute: $\int e^{8x} dx = \int e^u \frac{du}{8} = \frac{1}{8} \int e^u du$.
5. Integrate: $\frac{1}{8} e^u + C$.
6. Substitute back: $\frac{1}{8} e^{8x} + C$.

Example 2

Find the antiderivative of $4x e^{x^2}$.

1. Let $u = x^2$.
2. Find $\frac{du}{dx} = 2x$, so $du = 2x dx$.
3. Solve for $dx$: $dx = \frac{du}{2x}$.
4. Substitute: $\int 4x e^{x^2} dx = \int 4x e^u \frac{du}{2x} = 2 \int e^u du$.
5. Integrate: $2 e^u + C$.
6. Substitute back: $2 e^{x^2} + C$.

Antiderivatives of Rational Functions

Example 1

Find the antiderivative of $\frac{1}{x^2}$.

1. Rewrite: $\frac{1}{x^2} = x^{-2}$.
2. Apply the power rule: $\int x^{-2} dx = \frac{x^{-1}}{-1} + C = -\frac{1}{x} + C$

Example 2

Find the antiderivative of $\frac{1}{x^3}$.

1. Rewrite: $\frac{1}{x^3} = x^{-3}$.
2. Apply the power rule: $\int x^{-3} dx = \frac{x^{-2}}{-2} + C = -\frac{1}{2x^2} + C$

Example 3

Find the antiderivative of $\frac{8}{x^4}$.

1. Rewrite: $\frac{8}{x^4} = 8x^{-4}$.
2. Apply the power rule: $\int 8x^{-4} dx = 8 \frac{x^{-3}}{-3} + C = -\frac{8}{3x^3} + C$

Example 4

What is the antiderivative of $\frac{1}{(4x - 3)^2}$?

1. Let $u = 4x - 3$.
2. Then $\frac{du}{dx} = 4$, so $dx = \frac{du}{4}$.
3. Substituting gives: $\int \frac{1}{u^2} \frac{du}{4} = \frac{1}{4} \int u^{-2} du$.
4. Integrating: $\frac{1}{4} \frac{u^{-1}}{-1} + C = -\frac{1}{4u} + C$.
5. Substituting back gives: $\frac{-1}{4(4x - 3)} + C$

Example 5

What is the antiderivative of $\frac{7}{(5x - 3)^4}$?

1. Let $u = 5x - 3$.
2. Then $\frac{du}{dx} = 5$, so $dx = \frac{du}{5}$.
3. Substituting gives: $\int \frac{7}{u^4} \frac{du}{5} = \frac{7}{5} \int u^{-4} du$.
4. Integrating: $\frac{7}{5} \frac{u^{-3}}{-3} + C = -\frac{7}{15u^3} + C$.
5. Substituting back gives: $\frac{-7}{15(5x - 3)^3} + C$

Antiderivative of $\frac{1}{x}$

The antiderivative of $\frac{1}{x}$ is $\ln(x)$.

$\int \frac{1}{x} dx = \ln(x) + C$

Rationale

Since the derivative of $\ln(u)$ is $\frac{u'}{u}$, the derivative of $\ln(x)$ is $\frac{1}{x}$.

Example 1

Find the antiderivative of $\frac{7}{x}$:

$\int \frac{7}{x} dx = 7 \int \frac{1}{x} dx = 7 \ln(x) + C$

Example 2

Find the antiderivative of $\frac{1}{x + 5}$:

$\int \frac{1}{x + 5} dx = \ln(x + 5) + C$

Using u-substitution:

1. Let $u = x + 5$.
2. Then $du = dx$.
3. So, $\int \frac{1}{u} du = \ln(u) + C = \ln(x + 5) + C$