Calculus Examples

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

Step 1

Remove parentheses.

$\int 1 + 3 x^{2} - 5 x^{3} d x$

Step 2

Split the single integral into multiple integrals.

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

Step 3

Apply the constant rule.

$x + C + \int 3 x^{2} d x + \int - 5 x^{3} d x$

Step 4

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

$x + C + 3 \int x^{2} d x + \int - 5 x^{3} d x$

Step 5

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

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

Step 6

Since $- 5$ is constant with respect to $x$ , move $- 5$ out of the integral.

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

Step 7

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

$x + C + 3 (\frac{1}{3} x^{3} + C) - 5 (\frac{1}{4} x^{4} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$x + x^{3} - 5 (\frac{1}{4} x^{4}) + C$

Step 8.2

Simplify.

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Step 8.2.1

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

$x + x^{3} - 5 \frac{x^{4}}{4} + C$

Step 8.2.2

Combine $- 5$ and $\frac{x^{4}}{4}$ .

$x + x^{3} + \frac{- 5 x^{4}}{4} + C$

Step 8.2.3

Move the negative in front of the fraction.

$x + x^{3} - \frac{5 x^{4}}{4} + C$

Step 9

Reorder terms.

$x + x^{3} - \frac{5}{4} x^{4} + C$