Calculus Examples

$\int (3 x^{5} - x^{3} + 6) d x$

Step 1

Remove parentheses.

$\int 3 x^{5} - x^{3} + 6 d x$

Step 2

Split the single integral into multiple integrals.

$\int 3 x^{5} d x + \int - x^{3} d x + \int 6 d x$

Step 3

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

$3 \int x^{5} d x + \int - x^{3} d x + \int 6 d x$

Step 4

By the Power Rule, the integral of $x^{5}$ with respect to $x$ is $\frac{1}{6} x^{6}$ .

$3 (\frac{1}{6} x^{6} + C) + \int - x^{3} d x + \int 6 d x$

Step 5

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

$3 (\frac{1}{6} x^{6} + C) - \int x^{3} d x + \int 6 d x$

Step 6

By the Power Rule, the integral of $x^{3}$ with respect to $x$ is $\frac{1}{4} x^{4}$ .

$3 (\frac{1}{6} x^{6} + C) - (\frac{1}{4} x^{4} + C) + \int 6 d x$

Step 7

Apply the constant rule.

$3 (\frac{1}{6} x^{6} + C) - (\frac{1}{4} x^{4} + C) + 6 x + C$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{6}$ and $x^{6}$ .

$3 (\frac{x^{6}}{6} + C) - (\frac{1}{4} x^{4} + C) + 6 x + C$

Step 8.1.2

Combine $\frac{1}{4}$ and $x^{4}$ .

$3 (\frac{x^{6}}{6} + C) - (\frac{x^{4}}{4} + C) + 6 x + C$

Step 8.2

Simplify.

$\frac{x^{6}}{2} - \frac{x^{4}}{4} + 6 x + C$

Step 9

Reorder terms.

$\frac{1}{2} x^{6} - \frac{1}{4} x^{4} + 6 x + C$