Calculus Examples

$\int (4 + 3 x^{3} - 2 \sqrt{x}) d x$

Step 1

Remove parentheses.

$\int 4 + 3 x^{3} - 2 \sqrt{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int 4 d x + \int 3 x^{3} d x + \int - 2 \sqrt{x} d x$

Step 3

Apply the constant rule.

$4 x + C + \int 3 x^{3} d x + \int - 2 \sqrt{x} d x$

Step 4

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

$4 x + C + 3 \int x^{3} d x + \int - 2 \sqrt{x} d x$

Step 5

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

$4 x + C + 3 (\frac{1}{4} x^{4} + C) + \int - 2 \sqrt{x} d x$

Step 6

Since $- 2$ is constant with respect to $x$ , move $- 2$ out of the integral.

$4 x + C + 3 (\frac{1}{4} x^{4} + C) - 2 \int \sqrt{x} d x$

Step 7

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$4 x + C + 3 (\frac{1}{4} x^{4} + C) - 2 \int x^{\frac{1}{2}} d x$

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Simplify.

$4 x + \frac{3 x^{4}}{4} - 2 (\frac{2}{3} x^{\frac{3}{2}}) + C$

Step 9.2

Simplify.

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Step 9.2.1

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

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

Step 9.2.2

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

$4 x + \frac{3 x^{4}}{4} + \frac{- 2 (2 x^{\frac{3}{2}})}{3} + C$

Step 9.2.3

Multiply $2$ by $- 2$ .

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

Step 9.2.4

Move the negative in front of the fraction.

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

Step 10

Reorder terms.

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