Calculus Examples

$\int (- 3 x^{5} + 1 + 3 x^{3}) d x$

Step 1

Remove parentheses.

$\int - 3 x^{5} + 1 + 3 x^{3} d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Simplify.

$- \frac{x^{6}}{2} + x + 3 \frac{x^{4}}{4} + C$

Step 9.3

Combine $3$ and $\frac{x^{4}}{4}$ .

$- \frac{x^{6}}{2} + x + \frac{3 x^{4}}{4} + C$

Step 10

Reorder terms.

$- \frac{1}{2} x^{6} + x + \frac{3}{4} x^{4} + C$