Calculus Examples

$\int \frac{3 x (7^{x^{2}})}{7^{x^{2}} - 5} d x$

Step 1

Since $3$ is constant with respect to $x$ , move $3$ out of the integral.

$3 \int \frac{x \cdot (7^{x^{2}})}{7^{x^{2}} - 5} d x$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $x^{2}$ .

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

Step 2.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$2 x$

Step 2.2

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

$3 \int \frac{7^{u_{1}}}{7^{u_{1}} - 5} \cdot \frac{1}{2} d u_{1}$

Step 3

Simplify.

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Step 3.1

Multiply $\frac{7^{u_{1}}}{7^{u_{1}} - 5}$ by $\frac{1}{2}$ .

$3 \int \frac{7^{u_{1}}}{(7^{u_{1}} - 5) \cdot 2} d u_{1}$

Step 3.2

Move $2$ to the left of $7^{u_{1}} - 5$ .

$3 \int \frac{7^{u_{1}}}{2 (7^{u_{1}} - 5)} d u_{1}$

Step 4

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

$3 (\frac{1}{2} \int \frac{7^{u_{1}}}{7^{u_{1}} - 5} d u_{1})$

Step 5

Combine $\frac{1}{2}$ and $3$ .

$\frac{3}{2} \int \frac{7^{u_{1}}}{7^{u_{1}} - 5} d u_{1}$

Step 6

Let $u_{2} = 7^{u_{1}} - 5$ . Then $d u_{2} = 7^{u_{1}} \ln (7) d u_{1}$ , so $\frac{1}{\ln (7)} d u_{2} = 7^{u_{1}} d u_{1}$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

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Step 6.1

Let $u_{2} = 7^{u_{1}} - 5$ . Find $\frac{d u_{2}}{d u_{1}}$ .

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Step 6.1.1

Differentiate $7^{u_{1}} - 5$ .

$\frac{d}{d u_{1}} [7^{u_{1}} - 5]$

Step 6.1.2

By the Sum Rule, the derivative of $7^{u_{1}} - 5$ with respect to $u_{1}$ is $\frac{d}{d u_{1}} [7^{u_{1}}] + \frac{d}{d u_{1}} [- 5]$ .

$\frac{d}{d u_{1}} [7^{u_{1}}] + \frac{d}{d u_{1}} [- 5]$

Step 6.1.3

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $7$ .

$7^{u_{1}} \ln (7) + \frac{d}{d u_{1}} [- 5]$

Step 6.1.4

Differentiate using the Constant Rule.

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Step 6.1.4.1

Since $- 5$ is constant with respect to $u_{1}$ , the derivative of $- 5$ with respect to $u_{1}$ is $0$ .

$7^{u_{1}} \ln (7) + 0$

Step 6.1.4.2

Add $7^{u_{1}} \ln (7)$ and $0$ .

$7^{u_{1}} \ln (7)$

Step 6.2

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

$\frac{3}{2} \int \frac{1}{u_{2}} \cdot \frac{1}{\ln (7)} d u_{2}$

Step 7

Multiply $\frac{1}{u_{2}}$ by $\frac{1}{\ln (7)}$ .

$\frac{3}{2} \int \frac{1}{u_{2} \ln (7)} d u_{2}$

Step 8

Since $\frac{1}{\ln (7)}$ is constant with respect to $u_{2}$ , move $\frac{1}{\ln (7)}$ out of the integral.

$\frac{3}{2} (\frac{1}{\ln (7)} \int \frac{1}{u_{2}} d u_{2})$

Step 9

Simplify.

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Step 9.1

Multiply $\frac{1}{\ln (7)}$ by $\frac{3}{2}$ .

$\frac{3}{\ln (7) \cdot 2} \int \frac{1}{u_{2}} d u_{2}$

Step 9.2

Move $2$ to the left of $\ln (7)$ .

$\frac{3}{2 \ln (7)} \int \frac{1}{u_{2}} d u_{2}$

Step 10

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

$\frac{3}{2 \ln (7)} (\ln (| u_{2} |) + C)$

Step 11

Simplify.

$\frac{3}{2 \ln (7)} \ln (| u_{2} |) + C$

Step 12

Substitute back in for each integration substitution variable.

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Step 12.1

Replace all occurrences of $u_{2}$ with $7^{u_{1}} - 5$ .

$\frac{3}{2 \ln (7)} \ln (| 7^{u_{1}} - 5 |) + C$

Step 12.2

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

$\frac{3}{2 \ln (7)} \ln (| 7^{x^{2}} - 5 |) + C$