Calculus Examples

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

Step 1

Expand $x^{2} (1 - x^{3})$ .

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Step 1.1

Apply the distributive property.

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

Step 1.2

Reorder $x^{2}$ and $1$ .

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

Step 1.3

Reorder $x^{2}$ and $- 1$ .

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

Step 1.4

Multiply $x^{2}$ by $1$ .

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

Step 1.5

Factor out negative.

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

Step 1.6

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

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

Step 1.7

Add $2$ and $3$ .

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

Step 1.8

Reorder $x^{2}$ and $- x^{5}$ .

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

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

$- (\frac{1}{6} x^{6} + C) + \int x^{2} d x$

Step 5

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

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

Step 6

Simplify.

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Step 6.1

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

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

Step 6.2

Simplify.

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