Calculus Examples

$\int (x^{\frac{3}{2}} - 2 x^{\frac{2}{3}} + 5 \sqrt{x} - 3) d x$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

By the Power Rule, the integral of $x^{\frac{3}{2}}$ with respect to $x$ is $\frac{2}{5} x^{\frac{5}{2}}$ .

$\frac{2}{5} x^{\frac{5}{2}} + C + \int - 2 x^{\frac{2}{3}} d x + \int 5 \sqrt{x} d x + \int - 3 d x$

Step 4

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 2 \int x^{\frac{2}{3}} d x + \int 5 \sqrt{x} d x + \int - 3 d x$

Step 5

By the Power Rule, the integral of $x^{\frac{2}{3}}$ with respect to $x$ is $\frac{3}{5} x^{\frac{5}{3}}$ .

$\frac{2}{5} x^{\frac{5}{2}} + C - 2 (\frac{3}{5} x^{\frac{5}{3}} + C) + \int 5 \sqrt{x} d x + \int - 3 d x$

Step 6

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\frac{2}{5} x^{\frac{5}{2}} + C - 2 (\frac{3}{5} x^{\frac{5}{3}} + C) + 5 \int \sqrt{x} d x + \int - 3 d x$

Step 7

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 2 (\frac{3}{5} x^{\frac{5}{3}} + C) + 5 \int x^{\frac{1}{2}} d x + \int - 3 d x$

Step 8

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 2 (\frac{3}{5} x^{\frac{5}{3}} + C) + 5 (\frac{2}{3} x^{\frac{3}{2}} + C) + \int - 3 d x$

Step 9

Apply the constant rule.

$\frac{2}{5} x^{\frac{5}{2}} + C - 2 (\frac{3}{5} x^{\frac{5}{3}} + C) + 5 (\frac{2}{3} x^{\frac{3}{2}} + C) - 3 x + C$

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

Combine $\frac{3}{5}$ and $x^{\frac{5}{3}}$ .

$\frac{2}{5} x^{\frac{5}{2}} + C - 2 (\frac{3 x^{\frac{5}{3}}}{5} + C) + 5 (\frac{2}{3} x^{\frac{3}{2}} + C) - 3 x + C$

Step 10.1.2

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

$\frac{2}{5} x^{\frac{5}{2}} + C - 2 (\frac{3 x^{\frac{5}{3}}}{5} + C) + 5 (\frac{2 x^{\frac{3}{2}}}{3} + C) - 3 x + C$

Step 10.2

Simplify.

$\frac{2 x^{\frac{5}{2}}}{5} - \frac{6 x^{\frac{5}{3}}}{5} + \frac{10 x^{\frac{3}{2}}}{3} - 3 x + C$

Step 11

Reorder terms.

$\frac{2}{5} x^{\frac{5}{2}} - \frac{6}{5} x^{\frac{5}{3}} + \frac{10}{3} x^{\frac{3}{2}} - 3 x + C$