Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Multiply $3$ by $3$ .

$\int 9 x^{3} x^{2} d x$

Step 1.2

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

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Step 1.2.1

Move $x^{2}$ .

$\int 9 (x^{2} x^{3}) d x$

Step 1.2.2

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

$\int 9 x^{2 + 3} d x$

Step 1.2.3

Add $2$ and $3$ .

$\int 9 x^{5} d x$

Step 2

Since $9$ is constant with respect to $x$ , move $9$ out of the integral.

$9 \int x^{5} d x$

Step 3

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

$9 (\frac{1}{6} x^{6} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $9 (\frac{1}{6} x^{6} + C)$ as $9 (\frac{1}{6}) x^{6} + C$ .

$9 (\frac{1}{6}) x^{6} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $9$ and $\frac{1}{6}$ .

$\frac{9}{6} x^{6} + C$

Step 4.2.2

Cancel the common factor of $9$ and $6$ .

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Step 4.2.2.1

Factor $3$ out of $9$ .

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

Step 4.2.2.2

Cancel the common factors.

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Step 4.2.2.2.1

Factor $3$ out of $6$ .

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

Step 4.2.2.2.2

Cancel the common factor.

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

Step 4.2.2.2.3

Rewrite the expression.

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