Calculus Examples

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

Step 1

Simplify.

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Step 1.1

Raise $5$ to the power of $3$ .

$\int 125 \sqrt{x^{2}} d x$

Step 1.2

Pull terms out from under the radical, assuming positive real numbers.

$\int 125 x d x$

Step 2

Since $125$ is constant with respect to $x$ , move $125$ out of the integral.

$125 \int x d x$

Step 3

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

$125 (\frac{1}{2} x^{2} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $125 (\frac{1}{2} x^{2} + C)$ as $125 (\frac{1}{2}) x^{2} + C$ .

$125 (\frac{1}{2}) x^{2} + C$

Step 4.2

Combine $125$ and $\frac{1}{2}$ .

$\frac{125}{2} x^{2} + C$