Calculus Examples

$\int 4 x^{2} (x^{4} + 8) d x$

Step 1

Multiply $4 x^{2} (x^{4} + 8)$ .

$\int 4 x^{2} x^{4} + 4 x^{2} \cdot 8 d x$

Step 2

Simplify.

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Step 2.1

Multiply $x^{2}$ by $x^{4}$ by adding the exponents.

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Step 2.1.1

Move $x^{4}$ .

$\int 4 (x^{4} x^{2}) + 4 x^{2} \cdot 8 d x$

Step 2.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int 4 x^{4 + 2} + 4 x^{2} \cdot 8 d x$

Step 2.1.3

Add $4$ and $2$ .

$\int 4 x^{6} + 4 x^{2} \cdot 8 d x$

Step 2.2

Multiply $8$ by $4$ .

$\int 4 x^{6} + 32 x^{2} d x$

Step 3

Split the single integral into multiple integrals.

$\int 4 x^{6} d x + \int 32 x^{2} d x$

Step 4

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int x^{6} d x + \int 32 x^{2} d x$

Step 5

By the Power Rule, the integral of $x^{6}$ with respect to $x$ is $\frac{1}{7} x^{7}$ .

$4 (\frac{1}{7} x^{7} + C) + \int 32 x^{2} d x$

Step 6

Since $32$ is constant with respect to $x$ , move $32$ out of the integral.

$4 (\frac{1}{7} x^{7} + C) + 32 \int x^{2} d x$

Step 7

By the Power Rule, the integral of $x^{2}$ with respect to $x$ is $\frac{1}{3} x^{3}$ .

$4 (\frac{1}{7} x^{7} + C) + 32 (\frac{1}{3} x^{3} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$4 (\frac{1}{7}) x^{7} + 32 (\frac{1}{3}) x^{3} + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $4$ and $\frac{1}{7}$ .

$\frac{4}{7} x^{7} + 32 (\frac{1}{3}) x^{3} + C$

Step 8.2.2

Combine $32$ and $\frac{1}{3}$ .

$\frac{4}{7} x^{7} + \frac{32}{3} x^{3} + C$