Calculus Examples

$\int {(5 + 11 x^{2})}^{2} d x$

Step 1

Expand ${(5 + 11 x^{2})}^{2}$ .

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Step 1.1

Rewrite ${(5 + 11 x^{2})}^{2}$ as $(5 + 11 x^{2}) (5 + 11 x^{2})$ .

$\int (5 + 11 x^{2}) (5 + 11 x^{2}) d x$

Step 1.2

Apply the distributive property.

$\int 5 (5 + 11 x^{2}) + 11 x^{2} (5 + 11 x^{2}) d x$

Step 1.3

Apply the distributive property.

$\int 5 \cdot 5 + 5 (11 x^{2}) + 11 x^{2} (5 + 11 x^{2}) d x$

Step 1.4

Apply the distributive property.

$\int 5 \cdot 5 + 5 (11 x^{2}) + 11 x^{2} \cdot 5 + 11 x^{2} (11 x^{2}) d x$

Step 1.5

Move $x^{2}$ .

$\int 5 \cdot 5 + 5 \cdot 11 x^{2} + 11 \cdot 5 x^{2} + 11 x^{2} (11 x^{2}) d x$

Step 1.6

Move $x^{2}$ .

$\int 5 \cdot 5 + 5 \cdot 11 x^{2} + 11 \cdot 5 x^{2} + 11 \cdot 11 x^{2} x^{2} d x$

Step 1.7

Multiply $5$ by $5$ .

$\int 25 + 5 \cdot 11 x^{2} + 11 \cdot 5 x^{2} + 11 \cdot 11 x^{2} x^{2} d x$

Step 1.8

Multiply $5$ by $11$ .

$\int 25 + 55 x^{2} + 11 \cdot 5 x^{2} + 11 \cdot 11 x^{2} x^{2} d x$

Step 1.9

Multiply $11$ by $5$ .

$\int 25 + 55 x^{2} + 55 x^{2} + 11 \cdot 11 x^{2} x^{2} d x$

Step 1.10

Multiply $11$ by $11$ .

$\int 25 + 55 x^{2} + 55 x^{2} + 121 x^{2} x^{2} d x$

Step 1.11

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

$\int 25 + 55 x^{2} + 55 x^{2} + 121 x^{2 + 2} d x$

Step 1.12

Add $2$ and $2$ .

$\int 25 + 55 x^{2} + 55 x^{2} + 121 x^{4} d x$

Step 1.13

Add $55 x^{2}$ and $55 x^{2}$ .

$\int 25 + 110 x^{2} + 121 x^{4} d x$

Step 1.14

Reorder $110 x^{2}$ and $121 x^{4}$ .

$\int 25 + 121 x^{4} + 110 x^{2} d x$

Step 1.15

Move $25$ .

$\int 121 x^{4} + 110 x^{2} + 25 d x$

Step 2

Split the single integral into multiple integrals.

$\int 121 x^{4} d x + \int 110 x^{2} d x + \int 25 d x$

Step 3

Since $121$ is constant with respect to $x$ , move $121$ out of the integral.

$121 \int x^{4} d x + \int 110 x^{2} d x + \int 25 d x$

Step 4

By the Power Rule, the integral of $x^{4}$ with respect to $x$ is $\frac{1}{5} x^{5}$ .

$121 (\frac{1}{5} x^{5} + C) + \int 110 x^{2} d x + \int 25 d x$

Step 5

Since $110$ is constant with respect to $x$ , move $110$ out of the integral.

$121 (\frac{1}{5} x^{5} + C) + 110 \int x^{2} d x + \int 25 d x$

Step 6

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

$121 (\frac{1}{5} x^{5} + C) + 110 (\frac{1}{3} x^{3} + C) + \int 25 d x$

Step 7

Apply the constant rule.

$121 (\frac{1}{5} x^{5} + C) + 110 (\frac{1}{3} x^{3} + C) + 25 x + C$

Step 8

Simplify.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{1}{5}$ and $x^{5}$ .

$121 (\frac{x^{5}}{5} + C) + 110 (\frac{1}{3} x^{3} + C) + 25 x + C$

Step 8.1.2

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

$121 (\frac{x^{5}}{5} + C) + 110 (\frac{x^{3}}{3} + C) + 25 x + C$

Step 8.2

Simplify.

$\frac{121 x^{5}}{5} + \frac{110 x^{3}}{3} + 25 x + C$

Step 8.3

Reorder terms.

$\frac{121}{5} x^{5} + \frac{110}{3} x^{3} + 25 x + C$