Calculus Examples

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

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{3}$ .

$\frac{d}{d x} [x^{3}]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$3 x^{2}$

Step 1.2

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

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

Step 2

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

The integral of $5^{u_{1}}$ with respect to $u_{1}$ is $\frac{5^{u_{1}}}{\ln (5)}$ .

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

Step 5

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

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

Step 6

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

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Step 6.1

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

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Step 6.1.1

Differentiate $2 u_{1}$ .

$\frac{d}{d u_{1}} [2 u_{1}]$

Step 6.1.2

Since $2$ is constant with respect to $u_{1}$ , the derivative of $2 u_{1}$ with respect to $u_{1}$ is $2 \frac{d}{d u_{1}} [u_{1}]$ .

$2 \frac{d}{d u_{1}} [u_{1}]$

Step 6.1.3

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

$2 \cdot 1$

Step 6.1.4

Multiply $2$ by $1$ .

$2$

Step 6.2

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

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

Step 7

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

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

Step 8

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

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

Step 9

The integral of $3^{u_{2}}$ with respect to $u_{2}$ is $\frac{3^{u_{2}}}{\ln (3)}$ .

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

Step 10

Simplify.

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

Step 11

Substitute back in for each integration substitution variable.

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

$\frac{1}{3} \cdot \frac{5^{x^{3}}}{\ln (5)} + \frac{1}{3} (- \frac{3^{2 x^{3}}}{2 \ln (3)}) + C$

Step 12.2

Combine.

$\frac{1 \cdot 5^{x^{3}}}{3 \ln (5)} + \frac{1}{3} (- \frac{3^{2 x^{3}}}{2 \ln (3)}) + C$

Step 12.3

Multiply $\frac{1}{3} (- \frac{3^{2 x^{3}}}{2 \ln (3)})$ .

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Step 12.3.1

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

$\frac{1 \cdot 5^{x^{3}}}{3 \ln (5)} - \frac{3^{2 x^{3}}}{3 (2 \ln (3))} + C$

Step 12.3.2

Multiply $2$ by $3$ .

$\frac{1 \cdot 5^{x^{3}}}{3 \ln (5)} - \frac{3^{2 x^{3}}}{6 \ln (3)} + C$

Step 12.4

Multiply $5^{x^{3}}$ by $1$ .

$\frac{5^{x^{3}}}{3 \ln (5)} - \frac{3^{2 x^{3}}}{6 \ln (3)} + C$

Step 13

Reorder terms.

$\frac{1}{3 \ln (5)} \cdot 5^{x^{3}} - \frac{1}{6 \ln (3)} \cdot 3^{2 x^{3}} + C$