Calculus Examples

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

Step 1

Let $u = 1 - x^{2}$ . Then $d u = - 2 x d x$ , so $- \frac{1}{2} d u = x d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 1.1

Let $u = 1 - x^{2}$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $1 - x^{2}$ .

$\frac{d}{d x} [1 - x^{2}]$

Step 1.1.2

Differentiate.

Tap for more steps...

Step 1.1.2.1

By the Sum Rule, the derivative of $1 - x^{2}$ with respect to $x$ is $\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]$ .

$\frac{d}{d x} [1] + \frac{d}{d x} [- x^{2}]$

Step 1.1.2.2

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$0 + \frac{d}{d x} [- x^{2}]$

Step 1.1.3

Evaluate $\frac{d}{d x} [- x^{2}]$ .

Tap for more steps...

Step 1.1.3.1

Since $- 1$ is constant with respect to $x$ , the derivative of $- x^{2}$ with respect to $x$ is $- \frac{d}{d x} [x^{2}]$ .

$0 - \frac{d}{d x} [x^{2}]$

Step 1.1.3.2

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$0 - (2 x)$

Step 1.1.3.3

Multiply $2$ by $- 1$ .

$0 - 2 x$

Step 1.1.4

Subtract $2 x$ from $0$ .

$- 2 x$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int \frac{{\sqrt{- u + 1}}^{2}}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 2

Simplify.

Tap for more steps...

Step 2.1

Rewrite ${\sqrt{- u + 1}}^{2}$ as $- u + 1$ .

Tap for more steps...

Step 2.1.1

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

$\int \frac{{({(- u + 1)}^{\frac{1}{2}})}^{2}}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 2.1.2

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

$\int \frac{{(- u + 1)}^{\frac{1}{2} \cdot 2}}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 2.1.3

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

$\int \frac{{(- u + 1)}^{\frac{2}{2}}}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 2.1.4

Cancel the common factor of $2$ .

Tap for more steps...

Step 2.1.4.1

Cancel the common factor.

$\int \frac{{(- u + 1)}^{\frac{2}{2}}}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 2.1.4.2

Rewrite the expression.

$\int \frac{{(- u + 1)}^{1}}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 2.1.5

Simplify.

$\int \frac{- u + 1}{\sqrt{u}} \cdot \frac{1}{- 2} d u$

Step 2.2

Move the negative in front of the fraction.

$\int \frac{- u + 1}{\sqrt{u}} (- \frac{1}{2}) d u$

Step 2.3

Multiply $\frac{- u + 1}{\sqrt{u}}$ by $\frac{1}{2}$ .

$\int - \frac{- u + 1}{\sqrt{u} \cdot 2} d u$

Step 2.4

Move $2$ to the left of $\sqrt{u}$ .

$\int - \frac{- u + 1}{2 \sqrt{u}} d u$

Step 3

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

$- \int \frac{- u + 1}{2 \sqrt{u}} d u$

Step 4

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

$- (\frac{1}{2} \int \frac{- u + 1}{\sqrt{u}} d u)$

Step 5

Apply basic rules of exponents.

Tap for more steps...

Step 5.1

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

$- \frac{1}{2} \int \frac{- u + 1}{u^{\frac{1}{2}}} d u$

Step 5.2

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

$- \frac{1}{2} \int (- u + 1) {(u^{\frac{1}{2}})}^{- 1} d u$

Step 5.3

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

Tap for more steps...

Step 5.3.1

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

$- \frac{1}{2} \int (- u + 1) u^{\frac{1}{2} \cdot - 1} d u$

Step 5.3.2

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

$- \frac{1}{2} \int (- u + 1) u^{\frac{- 1}{2}} d u$

Step 5.3.3

Move the negative in front of the fraction.

$- \frac{1}{2} \int (- u + 1) u^{- \frac{1}{2}} d u$

Step 6

Expand $(- u + 1) u^{- \frac{1}{2}}$ .

Tap for more steps...

Step 6.1

Apply the distributive property.

$- \frac{1}{2} \int - u \cdot u^{- \frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 6.2

Factor out negative.

$- \frac{1}{2} \int - (u \cdot u^{- \frac{1}{2}}) + 1 u^{- \frac{1}{2}} d u$

Step 6.3

Raise $u$ to the power of $1$ .

$- \frac{1}{2} \int - (u^{1} u^{- \frac{1}{2}}) + 1 u^{- \frac{1}{2}} d u$

Step 6.4

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

$- \frac{1}{2} \int - u^{1 - \frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 6.5

Write $1$ as a fraction with a common denominator.

$- \frac{1}{2} \int - u^{\frac{2}{2} - \frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 6.6

Combine the numerators over the common denominator.

$- \frac{1}{2} \int - u^{\frac{2 - 1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 6.7

Subtract $1$ from $2$ .

$- \frac{1}{2} \int - u^{\frac{1}{2}} + 1 u^{- \frac{1}{2}} d u$

Step 6.8

Multiply $u^{- \frac{1}{2}}$ by $1$ .

$- \frac{1}{2} \int - u^{\frac{1}{2}} + u^{- \frac{1}{2}} d u$

Step 6.9

Reorder $- u^{\frac{1}{2}}$ and $u^{- \frac{1}{2}}$ .

$- \frac{1}{2} \int u^{- \frac{1}{2}} - u^{\frac{1}{2}} d u$

Step 7

Split the single integral into multiple integrals.

$- \frac{1}{2} (\int u^{- \frac{1}{2}} d u + \int - u^{\frac{1}{2}} d u)$

Step 8

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

$- \frac{1}{2} (2 u^{\frac{1}{2}} + C + \int - u^{\frac{1}{2}} d u)$

Step 9

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

$- \frac{1}{2} (2 u^{\frac{1}{2}} + C - \int u^{\frac{1}{2}} d u)$

Step 10

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

$- \frac{1}{2} (2 u^{\frac{1}{2}} + C - (\frac{2}{3} u^{\frac{3}{2}} + C))$

Step 11

Simplify.

$- \frac{1}{2} (2 u^{\frac{1}{2}} - \frac{2}{3} u^{\frac{3}{2}}) + C$

Step 12

Replace all occurrences of $u$ with $1 - x^{2}$ .

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

Step 13

Simplify.

Tap for more steps...

Step 13.1

Simplify each term.

Tap for more steps...

Step 13.1.1

Combine ${(1 - x^{2})}^{\frac{3}{2}}$ and $\frac{2}{3}$ .

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

Step 13.1.2

Move $2$ to the left of ${(1 - x^{2})}^{\frac{3}{2}}$ .

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

Step 13.2

To write $2 {(1 - x^{2})}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

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

Step 13.3

Combine $2 {(1 - x^{2})}^{\frac{1}{2}}$ and $\frac{3}{3}$ .

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

Step 13.4

Combine the numerators over the common denominator.

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

Step 13.5

Multiply $3$ by $2$ .

$- \frac{1}{2} \cdot \frac{6 {(1 - x^{2})}^{\frac{1}{2}} - 2 {(1 - x^{2})}^{\frac{3}{2}}}{3} + C$

Step 13.6

Multiply $- \frac{1}{2} \cdot \frac{6 {(1 - x^{2})}^{\frac{1}{2}} - 2 {(1 - x^{2})}^{\frac{3}{2}}}{3}$ .

Tap for more steps...

Step 13.6.1

Multiply $\frac{6 {(1 - x^{2})}^{\frac{1}{2}} - 2 {(1 - x^{2})}^{\frac{3}{2}}}{3}$ by $\frac{1}{2}$ .

$- \frac{6 {(1 - x^{2})}^{\frac{1}{2}} - 2 {(1 - x^{2})}^{\frac{3}{2}}}{3 \cdot 2} + C$

Step 13.6.2

Multiply $3$ by $2$ .

$- \frac{6 {(1 - x^{2})}^{\frac{1}{2}} - 2 {(1 - x^{2})}^{\frac{3}{2}}}{6} + C$

Step 14

Reorder terms.

$- \frac{1}{6} (6 {(1 - x^{2})}^{\frac{1}{2}} - 2 {(1 - x^{2})}^{\frac{3}{2}}) + C$