Calculus Examples

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

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x^{2}$ and $d v = \sqrt{1 - x}$ .

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

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Since $- \frac{2}{3} \cdot 2$ is constant with respect to $x$ , move $- \frac{2}{3} \cdot 2$ out of the integral.

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

Step 4

Simplify.

Tap for more steps...

Step 4.1

Multiply $2$ by $- 1$ .

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

Step 4.2

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

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

Step 4.3

Multiply $- 2$ by $2$ .

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

Step 4.4

Move the negative in front of the fraction.

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

Step 4.5

Multiply $- 1$ by $- 1$ .

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

Step 4.6

Multiply $\frac{4}{3}$ by $1$ .

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

Step 5

Let $u = 1 - x$ . Then $d u = - d x$ , so $- d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 5.1

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

Tap for more steps...

Step 5.1.1

Differentiate $1 - x$ .

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

Step 5.1.2

Differentiate.

Tap for more steps...

Step 5.1.2.1

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

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

Step 5.1.2.2

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

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

Step 5.1.3

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

Tap for more steps...

Step 5.1.3.1

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

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

Step 5.1.3.2

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

$0 - 1 \cdot 1$

Step 5.1.3.3

Multiply $- 1$ by $1$ .

$0 - 1$

Step 5.1.4

Subtract $1$ from $0$ .

$- 1$

Step 5.2

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

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

Step 6

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

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

Step 7

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

Tap for more steps...

Step 7.1

Apply the distributive property.

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

Step 7.2

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

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

Step 7.3

Reorder $u^{\frac{3}{2}}$ and $1$ .

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

Step 7.4

Factor out negative.

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

Step 7.5

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

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

Step 7.6

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

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

Step 7.7

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

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

Step 7.8

Combine the numerators over the common denominator.

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

Step 7.9

Add $3$ and $2$ .

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

Step 7.10

Multiply $u^{\frac{3}{2}}$ by $1$ .

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

Step 7.11

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

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

Step 8

Split the single integral into multiple integrals.

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

Step 9

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

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

Step 10

Combine $\frac{2}{5}$ and $u^{\frac{5}{2}}$ .

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

Step 11

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

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

Step 12

By the Power Rule, the integral of $u^{\frac{5}{2}}$ with respect to $u$ is $\frac{2}{7} u^{\frac{7}{2}}$ .

$- \frac{x^{2} (2 {(1 - x)}^{\frac{3}{2}})}{3} + \frac{4}{3} (- (\frac{2 u^{\frac{5}{2}}}{5} + C - (\frac{2}{7} u^{\frac{7}{2}} + C)))$

Step 13

Simplify.

Tap for more steps...

Step 13.1

Combine $\frac{2}{7}$ and $u^{\frac{7}{2}}$ .

$- \frac{x^{2} (2 {(1 - x)}^{\frac{3}{2}})}{3} + \frac{4}{3} (- (\frac{2 u^{\frac{5}{2}}}{5} + C - (\frac{2 u^{\frac{7}{2}}}{7} + C)))$

Step 13.2

Simplify.

$- \frac{2}{3} x^{2} {(1 - x)}^{\frac{3}{2}} + \frac{4}{3} (- (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{7} u^{\frac{7}{2}})) + C$

Step 13.3

Simplify.

Tap for more steps...

Step 13.3.1

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

$- \frac{x^{2} \cdot 2}{3} {(1 - x)}^{\frac{3}{2}} + \frac{4}{3} (- (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{7} u^{\frac{7}{2}})) + C$

Step 13.3.2

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

$- \frac{{(1 - x)}^{\frac{3}{2}} (x^{2} \cdot 2)}{3} + \frac{4}{3} (- (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{7} u^{\frac{7}{2}})) + C$

Step 13.3.3

Move $2$ to the left of $x^{2}$ .

$- \frac{{(1 - x)}^{\frac{3}{2}} (2 \cdot x^{2})}{3} + \frac{4}{3} (- (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{7} u^{\frac{7}{2}})) + C$

Step 13.3.4

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

$- \frac{2 \cdot {(1 - x)}^{\frac{3}{2}} x^{2}}{3} + \frac{4}{3} (- (\frac{2}{5} u^{\frac{5}{2}} - \frac{2}{7} u^{\frac{7}{2}})) + C$

Step 13.3.5

Combine $\frac{2}{5}$ and $u^{\frac{5}{2}}$ .

$- \frac{2 {(1 - x)}^{\frac{3}{2}} x^{2}}{3} + \frac{4}{3} (- (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2}{7} u^{\frac{7}{2}})) + C$

Step 13.3.6

Combine $u^{\frac{7}{2}}$ and $\frac{2}{7}$ .

$- \frac{2 {(1 - x)}^{\frac{3}{2}} x^{2}}{3} + \frac{4}{3} (- (\frac{2 u^{\frac{5}{2}}}{5} - \frac{u^{\frac{7}{2}} \cdot 2}{7})) + C$

Step 13.3.7

Move $2$ to the left of $u^{\frac{7}{2}}$ .

$- \frac{2 {(1 - x)}^{\frac{3}{2}} x^{2}}{3} + \frac{4}{3} (- (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 \cdot u^{\frac{7}{2}}}{7})) + C$

Step 13.3.8

To write $\frac{4}{3} (- (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7}))$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$- \frac{2 {(1 - x)}^{\frac{3}{2}} x^{2}}{3} + \frac{4}{3} (- (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7})) \cdot \frac{3}{3} + C$

Step 13.3.9

Combine $\frac{4}{3} (- (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7}))$ and $\frac{3}{3}$ .

$- \frac{2 {(1 - x)}^{\frac{3}{2}} x^{2}}{3} + \frac{\frac{4}{3} (- (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7})) \cdot 3}{3} + C$

Step 13.3.10

Combine the numerators over the common denominator.

$\frac{- 2 {(1 - x)}^{\frac{3}{2}} x^{2} + \frac{4}{3} (- (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7})) \cdot 3}{3} + C$

Step 13.3.11

Combine $\frac{4}{3}$ and $3$ .

$\frac{- 2 {(1 - x)}^{\frac{3}{2}} x^{2} - (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7}) \cdot \frac{4 \cdot 3}{3}}{3} + C$

Step 13.3.12

Multiply $4$ by $3$ .

$\frac{- 2 {(1 - x)}^{\frac{3}{2}} x^{2} - (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7}) \cdot \frac{12}{3}}{3} + C$

Step 13.3.13

Cancel the common factor of $12$ and $3$ .

Tap for more steps...

Step 13.3.13.1

Factor $3$ out of $12$ .

$\frac{- 2 {(1 - x)}^{\frac{3}{2}} x^{2} - (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7}) \cdot \frac{3 \cdot 4}{3}}{3} + C$

Step 13.3.13.2

Cancel the common factors.

Tap for more steps...

Step 13.3.13.2.1

Factor $3$ out of $3$ .

$\frac{- 2 {(1 - x)}^{\frac{3}{2}} x^{2} - (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7}) \cdot \frac{3 \cdot 4}{3 (1)}}{3} + C$

Step 13.3.13.2.2

Cancel the common factor.

$\frac{- 2 {(1 - x)}^{\frac{3}{2}} x^{2} - (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7}) \cdot \frac{3 \cdot 4}{3 \cdot 1}}{3} + C$

Step 13.3.13.2.3

Rewrite the expression.

$\frac{- 2 {(1 - x)}^{\frac{3}{2}} x^{2} - (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7}) \cdot \frac{4}{1}}{3} + C$

Step 13.3.13.2.4

Divide $4$ by $1$ .

$\frac{- 2 {(1 - x)}^{\frac{3}{2}} x^{2} - (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7}) \cdot 4}{3} + C$

Step 13.3.14

Multiply $4$ by $- 1$ .

$\frac{- 2 {(1 - x)}^{\frac{3}{2}} x^{2} - 4 (\frac{2 u^{\frac{5}{2}}}{5} - \frac{2 u^{\frac{7}{2}}}{7})}{3} + C$

Step 14

Replace all occurrences of $u$ with $1 - x$ .

$\frac{- 2 {(1 - x)}^{\frac{3}{2}} x^{2} - 4 (\frac{2 {(1 - x)}^{\frac{5}{2}}}{5} - \frac{2 {(1 - x)}^{\frac{7}{2}}}{7})}{3} + C$

Step 15

Reorder terms.

$\frac{1}{3} (- 2 {(1 - x)}^{\frac{3}{2}} x^{2} - 4 (\frac{2}{5} {(1 - x)}^{\frac{5}{2}} - \frac{2}{7} {(1 - x)}^{\frac{7}{2}})) + C$