Calculus Examples

$\int x^{4} - \frac{9}{2} \cdot \sqrt{x} + x^{- \frac{4}{3}} d x$

Step 1

Split the single integral into multiple integrals.

$\int x^{4} d x + \int - \frac{9}{2} \cdot \sqrt{x} d x + \int x^{- \frac{4}{3}} d x$

Step 2

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

$\frac{1}{5} x^{5} + C + \int - \frac{9}{2} \cdot \sqrt{x} d x + \int x^{- \frac{4}{3}} d x$

Step 3

Since $- \frac{9}{2}$ is constant with respect to $x$ , move $- \frac{9}{2}$ out of the integral.

$\frac{1}{5} x^{5} + C - \frac{9}{2} \int \sqrt{x} d x + \int x^{- \frac{4}{3}} d x$

Step 4

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

$\frac{1}{5} x^{5} + C - \frac{9}{2} \int x^{\frac{1}{2}} d x + \int x^{- \frac{4}{3}} d x$

Step 5

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

$\frac{1}{5} x^{5} + C - \frac{9}{2} (\frac{2}{3} x^{\frac{3}{2}} + C) + \int x^{- \frac{4}{3}} d x$

Step 6

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

$\frac{1}{5} x^{5} + C - \frac{9}{2} (\frac{2}{3} x^{\frac{3}{2}} + C) - 3 x^{- \frac{1}{3}} + C$

Step 7

Simplify.

Tap for more steps...

Step 7.1

Simplify.

$\frac{x^{5}}{5} - 3 x^{\frac{3}{2}} - 3 x^{- \frac{1}{3}} + C$

Step 7.2

Reorder terms.

$\frac{1}{5} x^{5} - 3 x^{\frac{3}{2}} - 3 x^{- \frac{1}{3}} + C$