Calculus Examples

$\int x (30 x^{2} + 90 x) d x$

Step 1

Simplify.

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Step 1.1

Apply the distributive property.

$\int x (30 x^{2}) + x (90 x) d x$

Step 1.2

Reorder $x$ and $30$ .

$\int 30 \cdot x \cdot x^{2} + x (90 x) d x$

Step 1.3

Reorder $x$ and $90$ .

$\int 30 \cdot x \cdot x^{2} + 90 \cdot x \cdot x d x$

Step 1.4

Raise $x$ to the power of $1$ .

$\int 30 (x^{1} x^{2}) + 90 x \cdot x d x$

Step 1.5

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

$\int 30 x^{1 + 2} + 90 x \cdot x d x$

Step 1.6

Add $1$ and $2$ .

$\int 30 x^{3} + 90 x \cdot x d x$

Step 1.7

Raise $x$ to the power of $1$ .

$\int 30 x^{3} + 90 (x^{1} x) d x$

Step 1.8

Raise $x$ to the power of $1$ .

$\int 30 x^{3} + 90 (x^{1} x^{1}) d x$

Step 1.9

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

$\int 30 x^{3} + 90 x^{1 + 1} d x$

Step 1.10

Add $1$ and $1$ .

$\int 30 x^{3} + 90 x^{2} d x$

Step 2

Split the single integral into multiple integrals.

$\int 30 x^{3} d x + \int 90 x^{2} d x$

Step 3

Since $30$ is constant with respect to $x$ , move $30$ out of the integral.

$30 \int x^{3} d x + \int 90 x^{2} d x$

Step 4

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

$30 (\frac{1}{4} x^{4} + C) + \int 90 x^{2} d x$

Step 5

Since $90$ is constant with respect to $x$ , move $90$ out of the integral.

$30 (\frac{1}{4} x^{4} + C) + 90 \int x^{2} d x$

Step 6

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

$30 (\frac{1}{4} x^{4} + C) + 90 (\frac{1}{3} x^{3} + C)$

Step 7

Simplify.

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Step 7.1

Simplify.

$30 (\frac{1}{4}) x^{4} + 90 (\frac{1}{3}) x^{3} + C$

Step 7.2

Simplify.

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Step 7.2.1

Combine $30$ and $\frac{1}{4}$ .

$\frac{30}{4} x^{4} + 90 (\frac{1}{3}) x^{3} + C$

Step 7.2.2

Cancel the common factor of $30$ and $4$ .

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Step 7.2.2.1

Factor $2$ out of $30$ .

$\frac{2 (15)}{4} x^{4} + 90 (\frac{1}{3}) x^{3} + C$

Step 7.2.2.2

Cancel the common factors.

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Step 7.2.2.2.1

Factor $2$ out of $4$ .

$\frac{2 \cdot 15}{2 \cdot 2} x^{4} + 90 (\frac{1}{3}) x^{3} + C$

Step 7.2.2.2.2

Cancel the common factor.

$\frac{2 \cdot 15}{2 \cdot 2} x^{4} + 90 (\frac{1}{3}) x^{3} + C$

Step 7.2.2.2.3

Rewrite the expression.

$\frac{15}{2} x^{4} + 90 (\frac{1}{3}) x^{3} + C$

Step 7.2.3

Combine $90$ and $\frac{1}{3}$ .

$\frac{15}{2} x^{4} + \frac{90}{3} x^{3} + C$

Step 7.2.4

Cancel the common factor of $90$ and $3$ .

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Step 7.2.4.1

Factor $3$ out of $90$ .

$\frac{15}{2} x^{4} + \frac{3 \cdot 30}{3} x^{3} + C$

Step 7.2.4.2

Cancel the common factors.

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Step 7.2.4.2.1

Factor $3$ out of $3$ .

$\frac{15}{2} x^{4} + \frac{3 \cdot 30}{3 (1)} x^{3} + C$

Step 7.2.4.2.2

Cancel the common factor.

$\frac{15}{2} x^{4} + \frac{3 \cdot 30}{3 \cdot 1} x^{3} + C$

Step 7.2.4.2.3

Rewrite the expression.

$\frac{15}{2} x^{4} + \frac{30}{1} x^{3} + C$

Step 7.2.4.2.4

Divide $30$ by $1$ .

$\frac{15}{2} x^{4} + 30 x^{3} + C$