Calculus Examples

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

Step 1

This integral could not be completed using u-substitution. Mathway will use another method.

Step 2

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

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

Step 3

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

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

Step 4

Simplify the answer.

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Step 4.1

Rewrite $3 (\frac{1}{\sqrt{3} + 1} x^{\sqrt{3} + 1} + C)$ as $3 (\frac{\sqrt{3} - 1}{2} x^{\sqrt{3} + 1}) + C$ .

$3 (\frac{\sqrt{3} - 1}{2} x^{\sqrt{3} + 1}) + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $\frac{\sqrt{3} - 1}{2}$ and $x^{\sqrt{3} + 1}$ .

$3 \frac{(\sqrt{3} - 1) x^{\sqrt{3} + 1}}{2} + C$

Step 4.2.2

Combine $3$ and $\frac{(\sqrt{3} - 1) x^{\sqrt{3} + 1}}{2}$ .

$\frac{3 (\sqrt{3} - 1) x^{\sqrt{3} + 1}}{2} + C$

Step 4.3

Simplify.

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Step 4.3.1

Apply the distributive property.

$\frac{(3 \sqrt{3} + 3 \cdot - 1) x^{\sqrt{3} + 1}}{2} + C$

Step 4.3.2

Multiply $3$ by $- 1$ .

$\frac{(3 \sqrt{3} - 3) x^{\sqrt{3} + 1}}{2} + C$