Calculus Examples

$\int x^{4} - 3 \sqrt{x} + x^{- \frac{5}{4}} d x$

Step 1

Split the single integral into multiple integrals.

$\int x^{4} d x + \int - 3 \sqrt{x} d x + \int x^{- \frac{5}{4}} d x$

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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Step 7.1

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

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

Step 7.2

Simplify.

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

Step 7.3

Reorder terms.

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