Calculus Examples

$\int \sqrt{3 + s} {(s + 1)}^{2} d s$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = {(s + 1)}^{2}$ and $d v = \sqrt{3 + s}$ .

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

Step 2

Simplify.

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Step 2.1

Combine $\frac{2}{3}$ and ${(3 + s)}^{\frac{3}{2}}$ .

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

Step 2.2

Combine ${(s + 1)}^{2}$ and $\frac{2 {(3 + s)}^{\frac{3}{2}}}{3}$ .

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

Step 2.3

Move $2$ to the left of ${(s + 1)}^{2}$ .

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

Step 3

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

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

Step 4

Let $u = 3 + s$ . Then $d u = d s$ . Rewrite using $u$ and $d$ $u$ .

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Step 4.1

Let $u = 3 + s$ . Find $\frac{d u}{d s}$ .

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Step 4.1.1

Differentiate $3 + s$ .

$\frac{d}{d s} [3 + s]$

Step 4.1.2

By the Sum Rule, the derivative of $3 + s$ with respect to $s$ is $\frac{d}{d s} [3] + \frac{d}{d s} [s]$ .

$\frac{d}{d s} [3] + \frac{d}{d s} [s]$

Step 4.1.3

Since $3$ is constant with respect to $s$ , the derivative of $3$ with respect to $s$ is $0$ .

$0 + \frac{d}{d s} [s]$

Step 4.1.4

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

$0 + 1$

Step 4.1.5

Add $0$ and $1$ .

$1$

Step 4.2

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

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

Step 5

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

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Step 5.1

Apply the distributive property.

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

Step 5.2

Apply the distributive property.

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

Step 5.3

Apply the distributive property.

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

Step 5.4

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

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

Step 5.5

Move $u^{\frac{3}{2}}$ .

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

Step 5.6

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

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

Step 5.7

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

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

Step 5.8

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

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

Step 5.9

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

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

Step 5.10

Combine the numerators over the common denominator.

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

Step 5.11

Add $3$ and $2$ .

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

Step 5.12

Multiply $2$ by $- 3$ .

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}}}{3} - \frac{2}{3} \int 2 u^{\frac{5}{2}} - 6 u^{\frac{3}{2}} + 2 u^{\frac{3}{2}} d u$

Step 5.13

Add $- 6 u^{\frac{3}{2}}$ and $2 u^{\frac{3}{2}}$ .

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

Since $2$ is constant with respect to $u$ , move $2$ out of the integral.

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

Step 8

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

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}}}{3} - \frac{2}{3} (2 (\frac{2}{7} u^{\frac{7}{2}} + C) + \int - 4 u^{\frac{3}{2}} d u)$

Step 9

Since $- 4$ is constant with respect to $u$ , move $- 4$ out of the integral.

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}}}{3} - \frac{2}{3} (2 (\frac{2}{7} u^{\frac{7}{2}} + C) - 4 \int u^{\frac{3}{2}} d u)$

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Simplify.

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}}}{3} - \frac{2}{3} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5}) + C$

Step 11.2

Simplify.

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Step 11.2.1

To write $- \frac{2}{3} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}}}{3} - \frac{2}{3} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5}) \cdot \frac{3}{3} + C$

Step 11.2.2

Combine $- \frac{2}{3} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5})$ and $\frac{3}{3}$ .

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}}}{3} + \frac{- \frac{2}{3} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5}) \cdot 3}{3} + C$

Step 11.2.3

Combine the numerators over the common denominator.

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} - \frac{2}{3} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5}) \cdot 3}{3} + C$

Step 11.2.4

Multiply $3$ by $- 1$ .

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} - 3 (\frac{2}{3}) (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5})}{3} + C$

Step 11.2.5

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

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} + \frac{- 3 \cdot 2}{3} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5})}{3} + C$

Step 11.2.6

Multiply $- 3$ by $2$ .

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} + \frac{- 6}{3} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5})}{3} + C$

Step 11.2.7

Cancel the common factor of $- 6$ and $3$ .

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Step 11.2.7.1

Factor $3$ out of $- 6$ .

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} + \frac{3 \cdot - 2}{3} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5})}{3} + C$

Step 11.2.7.2

Cancel the common factors.

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Step 11.2.7.2.1

Factor $3$ out of $3$ .

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} + \frac{3 \cdot - 2}{3 (1)} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5})}{3} + C$

Step 11.2.7.2.2

Cancel the common factor.

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} + \frac{3 \cdot - 2}{3 \cdot 1} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5})}{3} + C$

Step 11.2.7.2.3

Rewrite the expression.

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} + \frac{- 2}{1} (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5})}{3} + C$

Step 11.2.7.2.4

Divide $- 2$ by $1$ .

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} - 2 (\frac{4 u^{\frac{7}{2}}}{7} - \frac{8 u^{\frac{5}{2}}}{5})}{3} + C$

Step 12

Replace all occurrences of $u$ with $3 + s$ .

$\frac{2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} - 2 (\frac{4 {(3 + s)}^{\frac{7}{2}}}{7} - \frac{8 {(3 + s)}^{\frac{5}{2}}}{5})}{3} + C$

Step 13

Reorder terms.

$\frac{1}{3} (2 {(s + 1)}^{2} {(3 + s)}^{\frac{3}{2}} - 2 (\frac{4}{7} {(3 + s)}^{\frac{7}{2}} - \frac{8}{5} {(3 + s)}^{\frac{5}{2}})) + C$