Calculus Examples

$\int x^{4} (x^{3} + 8 x^{2} + 7) d x$

Step 1

Expand $x^{4} (x^{3} + 8 x^{2} + 7)$ .

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Step 1.1

Apply the distributive property.

$\int x^{4} (x^{3} + 8 x^{2}) + x^{4} \cdot 7 d x$

Step 1.2

Apply the distributive property.

$\int x^{4} x^{3} + x^{4} (8 x^{2}) + x^{4} \cdot 7 d x$

Step 1.3

Reorder $x^{4}$ and $8$ .

$\int x^{4} x^{3} + 8 \cdot x^{4} x^{2} + x^{4} \cdot 7 d x$

Step 1.4

Reorder $x^{4}$ and $7$ .

$\int x^{4} x^{3} + 8 \cdot x^{4} x^{2} + 7 \cdot x^{4} d x$

Step 1.5

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

$\int x^{4 + 3} + 8 x^{4} x^{2} + 7 x^{4} d x$

Step 1.6

Add $4$ and $3$ .

$\int x^{7} + 8 x^{4} x^{2} + 7 x^{4} d x$

Step 1.7

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

$\int x^{7} + 8 x^{4 + 2} + 7 x^{4} d x$

Step 1.8

Add $4$ and $2$ .

$\int x^{7} + 8 x^{6} + 7 x^{4} d x$

Step 2

Split the single integral into multiple integrals.

$\int x^{7} d x + \int 8 x^{6} d x + \int 7 x^{4} d x$

Step 3

By the Power Rule, the integral of $x^{7}$ with respect to $x$ is $\frac{1}{8} x^{8}$ .

$\frac{1}{8} x^{8} + C + \int 8 x^{6} d x + \int 7 x^{4} d x$

Step 4

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$\frac{1}{8} x^{8} + C + 8 \int x^{6} d x + \int 7 x^{4} d x$

Step 5

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

$\frac{1}{8} x^{8} + C + 8 (\frac{1}{7} x^{7} + C) + \int 7 x^{4} d x$

Step 6

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$\frac{1}{8} x^{8} + C + 8 (\frac{1}{7} x^{7} + C) + 7 \int x^{4} d x$

Step 7

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$\frac{1}{8} x^{8} + C + 8 (\frac{1}{7} x^{7} + C) + 7 (\frac{1}{5} x^{5} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{x^{8}}{8} + \frac{8 x^{7}}{7} + 7 (\frac{1}{5} x^{5}) + C$

Step 8.2

Simplify.

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Step 8.2.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$\frac{x^{8}}{8} + \frac{8 x^{7}}{7} + 7 \frac{x^{5}}{5} + C$

Step 8.2.2

Combine $7$ and $\frac{x^{5}}{5}$ .

$\frac{x^{8}}{8} + \frac{8 x^{7}}{7} + \frac{7 x^{5}}{5} + C$

Step 8.3

Reorder terms.

$\frac{1}{8} x^{8} + \frac{8}{7} x^{7} + \frac{7}{5} x^{5} + C$