Calculus Examples

$\int \sqrt[3]{x^{2}} - \frac{3}{\sqrt{x}} + 7 d x$

Step 1

Split the single integral into multiple integrals.

$\int \sqrt[3]{x^{2}} d x + \int - \frac{3}{\sqrt{x}} d x + \int 7 d x$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ .

$\int x^{\frac{2}{3}} d x + \int - \frac{3}{\sqrt{x}} d x + \int 7 d x$

Step 3

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

$\frac{3}{5} x^{\frac{5}{3}} + C + \int - \frac{3}{\sqrt{x}} d x + \int 7 d x$

Step 4

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$\frac{3}{5} x^{\frac{5}{3}} + C - \int \frac{3}{\sqrt{x}} d x + \int 7 d x$

Step 5

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

$\frac{3}{5} x^{\frac{5}{3}} + C - (3 \int \frac{1}{\sqrt{x}} d x) + \int 7 d x$

Step 6

Simplify the expression.

Tap for more steps...

Step 6.1

Multiply $3$ by $- 1$ .

$\frac{3}{5} x^{\frac{5}{3}} + C - 3 \int \frac{1}{\sqrt{x}} d x + \int 7 d x$

Step 6.2

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

$\frac{3}{5} x^{\frac{5}{3}} + C - 3 \int \frac{1}{x^{\frac{1}{2}}} d x + \int 7 d x$

Step 6.3

Move $x^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\frac{3}{5} x^{\frac{5}{3}} + C - 3 \int {(x^{\frac{1}{2}})}^{- 1} d x + \int 7 d x$

Step 6.4

Multiply the exponents in ${(x^{\frac{1}{2}})}^{- 1}$ .

Tap for more steps...

Step 6.4.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\frac{3}{5} x^{\frac{5}{3}} + C - 3 \int x^{\frac{1}{2} \cdot - 1} d x + \int 7 d x$

Step 6.4.2

Combine $\frac{1}{2}$ and $- 1$ .

$\frac{3}{5} x^{\frac{5}{3}} + C - 3 \int x^{\frac{- 1}{2}} d x + \int 7 d x$

Step 6.4.3

Move the negative in front of the fraction.

$\frac{3}{5} x^{\frac{5}{3}} + C - 3 \int x^{- \frac{1}{2}} d x + \int 7 d x$

Step 7

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

$\frac{3}{5} x^{\frac{5}{3}} + C - 3 (2 x^{\frac{1}{2}} + C) + \int 7 d x$

Step 8

Apply the constant rule.

$\frac{3}{5} x^{\frac{5}{3}} + C - 3 (2 x^{\frac{1}{2}} + C) + 7 x + C$

Step 9

Simplify.

Tap for more steps...

Step 9.1

Simplify.

$\frac{3 x^{\frac{5}{3}}}{5} - 6 x^{\frac{1}{2}} + 7 x + C$

Step 9.2

Reorder terms.

$\frac{3}{5} x^{\frac{5}{3}} - 6 x^{\frac{1}{2}} + 7 x + C$