Calculus Examples

$\int \frac{3^{2 x}}{1 + 3^{2 x}} d x$

Step 1

Let $u_{1} = 2 x$ . Then $d u_{1} = 2 d x$ , so $\frac{1}{2} d u_{1} = d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 1.1

Let $u_{1} = 2 x$ . Find $\frac{d u_{1}}{d x}$ .

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Step 1.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 1.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 1.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 1.1.4

Multiply $2$ by $1$ .

$2$

Step 1.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$\int \frac{3^{u_{1}}}{1 + 3^{u_{1}}} \cdot \frac{1}{2} d u_{1}$

Step 2

Simplify.

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Step 2.1

Multiply $\frac{3^{u_{1}}}{1 + 3^{u_{1}}}$ by $\frac{1}{2}$ .

$\int \frac{3^{u_{1}}}{(1 + 3^{u_{1}}) \cdot 2} d u_{1}$

Step 2.2

Move $2$ to the left of $1 + 3^{u_{1}}$ .

$\int \frac{3^{u_{1}}}{2 (1 + 3^{u_{1}})} d u_{1}$

Step 3

Since $\frac{1}{2}$ is constant with respect to $u_{1}$ , move $\frac{1}{2}$ out of the integral.

$\frac{1}{2} \int \frac{3^{u_{1}}}{1 + 3^{u_{1}}} d u_{1}$

Step 4

Let $u_{2} = 1 + 3^{u_{1}}$ . Then $d u_{2} = 3^{u_{1}} \ln (3) d u_{1}$ , so $\frac{1}{\ln (3)} d u_{2} = 3^{u_{1}} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 4.1

Let $u_{2} = 1 + 3^{u_{1}}$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 4.1.1

Differentiate $1 + 3^{u_{1}}$ .

$\frac{d}{d u_{1}} [1 + 3^{u_{1}}]$

Step 4.1.2

Differentiate.

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Step 4.1.2.1

By the Sum Rule, the derivative of $1 + 3^{u_{1}}$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [1] + \frac{d}{d u_{1}} [3^{u_{1}}]$ .

$\frac{d}{d u_{1}} [1] + \frac{d}{d u_{1}} [3^{u_{1}}]$

Step 4.1.2.2

Since $1$ is constant with respect to $u_{1}$ , the derivative of $1$ with respect to $u_{1}$ is $0$ .

$0 + \frac{d}{d u_{1}} [3^{u_{1}}]$

Step 4.1.3

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $3$ .

$0 + 3^{u_{1}} \ln (3)$

Step 4.1.4

Add $0$ and $3^{u_{1}} \ln (3)$ .

$3^{u_{1}} \ln (3)$

Step 4.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\frac{1}{2} \int \frac{1}{u_{2}} \cdot \frac{1}{\ln (3)} d u_{2}$

Step 5

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{\ln (3)}$ .

$\frac{1}{2} \int \frac{1}{u_{2} \ln (3)} d u_{2}$

Step 6

Since $\frac{1}{\ln (3)}$ is constant with respect to $u_{2}$ , move $\frac{1}{\ln (3)}$ out of the integral.

$\frac{1}{2} (\frac{1}{\ln (3)} \int \frac{1}{u_{2}} d u_{2})$

Step 7

Simplify.

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Step 7.1

Multiply $\frac{1}{\ln (3)}$ by $\frac{1}{2}$ .

$\frac{1}{\ln (3) \cdot 2} \int \frac{1}{u_{2}} d u_{2}$

Step 7.2

Move $2$ to the left of $\ln (3)$ .

$\frac{1}{2 \ln (3)} \int \frac{1}{u_{2}} d u_{2}$

Step 8

The integral of $\frac{1}{u_{2}}$ with respect to $u_{2}$ is $\ln (| u_{2} |)$ .

$\frac{1}{2 \ln (3)} (\ln (| u_{2} |) + C)$

Step 9

Simplify.

$\frac{1}{2 \ln (3)} \ln (| u_{2} |) + C$

Step 10

Substitute back in for each integration substitution variable.

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Step 10.1

Replace all occurrences of $u_{2}$ with $1 + 3^{u_{1}}$ .

$\frac{1}{2 \ln (3)} \ln (| 1 + 3^{u_{1}} |) + C$

Step 10.2

Replace all occurrences of $u_{1}$ with $2 x$ .

$\frac{1}{2 \ln (3)} \ln (| 1 + 3^{2 x} |) + C$