Calculus Examples

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

Step 1

Multiply $(x^{3} + 3) x^{2}$ .

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

Step 2

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

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Step 2.1

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

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

Step 2.2

Add $3$ and $2$ .

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

$\frac{1}{6} x^{6} + C + \int 3 x^{2} d x$

Step 5

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

$\frac{1}{6} x^{6} + C + 3 \int x^{2} d x$

Step 6

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

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

Step 7

Simplify.

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Step 7.1

Simplify.

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

Step 7.2

Simplify.

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Step 7.2.1

Combine $3$ and $\frac{1}{3}$ .

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

Step 7.2.2

Cancel the common factor of $3$ .

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Step 7.2.2.1

Cancel the common factor.

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

Step 7.2.2.2

Rewrite the expression.

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

Step 7.2.3

Multiply $x^{3}$ by $1$ .

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