Calculus Examples

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

Step 1

Simplify.

Tap for more steps...

Step 1.1

Simplify the numerator.

Tap for more steps...

Step 1.1.1

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\int \frac{\sqrt{x} - \frac{1}{x^{3}}}{x^{2}} d x$

Step 1.1.2

To write $\sqrt{x}$ as a fraction with a common denominator, multiply by $\frac{x^{3}}{x^{3}}$ .

$\int \frac{\frac{\sqrt{x} x^{3}}{x^{3}} - \frac{1}{x^{3}}}{x^{2}} d x$

Step 1.1.3

Combine the numerators over the common denominator.

$\int \frac{\frac{\sqrt{x} x^{3} - 1}{x^{3}}}{x^{2}} d x$

Step 1.2

Multiply the numerator by the reciprocal of the denominator.

$\int \frac{\sqrt{x} x^{3} - 1}{x^{3}} \cdot \frac{1}{x^{2}} d x$

Step 1.3

Multiply $\frac{\sqrt{x} x^{3} - 1}{x^{3}} \cdot \frac{1}{x^{2}}$ .

Tap for more steps...

Step 1.3.1

Multiply $\frac{\sqrt{x} x^{3} - 1}{x^{3}}$ by $\frac{1}{x^{2}}$ .

$\int \frac{\sqrt{x} x^{3} - 1}{x^{3} x^{2}} d x$

Step 1.3.2

Multiply $x^{3}$ by $x^{2}$ by adding the exponents.

Tap for more steps...

Step 1.3.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int \frac{\sqrt{x} x^{3} - 1}{x^{3 + 2}} d x$

Step 1.3.2.2

Add $3$ and $2$ .

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

Multiply $x^{\frac{1}{2}}$ by $x^{3}$ by adding the exponents.

Tap for more steps...

Step 2.2.1

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 2.2.2

To write $3$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 2.2.3

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

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

Step 2.2.4

Combine the numerators over the common denominator.

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

Step 2.2.5

Simplify the numerator.

Tap for more steps...

Step 2.2.5.1

Multiply $3$ by $2$ .

$\int \frac{x^{\frac{1 + 6}{2}} - 1}{x^{5}} d x$

Step 2.2.5.2

Add $1$ and $6$ .

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

Step 3

Apply basic rules of exponents.

Tap for more steps...

Step 3.1

Move $x^{5}$ out of the denominator by raising it to the $- 1$ power.

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

Step 3.2

Multiply the exponents in ${(x^{5})}^{- 1}$ .

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Multiply $5$ by $- 1$ .

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

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

Step 4.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 4.3

To write $- 5$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 4.4

Combine $- 5$ and $\frac{2}{2}$ .

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

Tap for more steps...

Step 4.6.1

Multiply $- 5$ by $2$ .

$\int x^{\frac{7 - 10}{2}} - 1 x^{- 5} d x$

Step 4.6.2

Subtract $10$ from $7$ .

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

Step 4.7

Reorder $x^{\frac{- 3}{2}}$ and $- 1 x^{- 5}$ .

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

Step 5

Simplify.

Tap for more steps...

Step 5.1

Rewrite $- 1 x^{- 5}$ as $- x^{- 5}$ .

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

Step 5.2

Move the negative in front of the fraction.

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

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

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

Step 8

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

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

Step 9

Simplify.

Tap for more steps...

Step 9.1

Combine $x^{- 4}$ and $\frac{1}{4}$ .

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

Step 9.2

Move $x^{- 4}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 10

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

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

Step 11

Simplify.

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