Calculus Examples

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

Step 1

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

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

Step 2

Simplify.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

$(t^{2} + 3 t + 2) \frac{2 t^{\frac{3}{2}}}{3} - \int \frac{2 t^{\frac{3}{2}}}{3} (2 t) + \frac{2 t^{\frac{3}{2}}}{3} \cdot 3 d t$

Step 3.2

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

$(t^{2} + 3 t + 2) \frac{2 t^{\frac{3}{2}}}{3} - \int 2 \cdot \frac{2 t^{\frac{3}{2}}}{3} t + \frac{2 t^{\frac{3}{2}}}{3} \cdot 3 d t$

Step 3.3

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

$(t^{2} + 3 t + 2) \frac{2 t^{\frac{3}{2}}}{3} - \int 2 \cdot \frac{2 t^{\frac{3}{2}}}{3} t + 3 \cdot \frac{2 t^{\frac{3}{2}}}{3} d t$

Step 3.4

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

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

Step 3.5

Multiply $2$ by $2$ .

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

Step 3.6

Combine $\frac{4 t^{\frac{3}{2}}}{3}$ and $t$ .

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

Step 3.7

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

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

Step 3.8

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

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

Step 3.9

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

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

Step 3.10

Combine the numerators over the common denominator.

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

Step 3.11

Add $3$ and $2$ .

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

Step 3.12

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

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

Step 3.13

Multiply $3$ by $2$ .

$(t^{2} + 3 t + 2) \frac{2 t^{\frac{3}{2}}}{3} - \int \frac{4 t^{\frac{5}{2}}}{3} + \frac{6 t^{\frac{3}{2}}}{3} d t$

Step 4

Simplify.

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Step 4.1

Factor $3$ out of $6 t^{\frac{3}{2}}$ .

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

Step 4.2

Cancel the common factors.

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Step 4.2.1

Factor $3$ out of $3$ .

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

Step 4.2.2

Cancel the common factor.

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

Step 4.2.3

Rewrite the expression.

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

Step 4.2.4

Divide $2 t^{\frac{3}{2}}$ by $1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Since $\frac{4}{3}$ is constant with respect to $t$ , move $\frac{4}{3}$ out of the integral.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

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

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

Step 11.2

Simplify.

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

Step 11.3

Simplify.

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Step 11.3.1

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

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

Step 11.3.2

Multiply $\frac{4}{3}$ by $\frac{2 t^{\frac{7}{2}}}{7}$ .

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

Step 11.3.3

Multiply $2$ by $4$ .

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

Step 11.3.4

Multiply $3$ by $7$ .

$\frac{2 t^{\frac{7}{2}}}{3} + 2 t^{\frac{5}{2}} + \frac{4 t^{\frac{3}{2}}}{3} - (\frac{8 t^{\frac{7}{2}}}{21} + 2 \frac{2 t^{\frac{5}{2}}}{5}) + C$

Step 11.3.5

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

$\frac{2 t^{\frac{7}{2}}}{3} + 2 t^{\frac{5}{2}} + \frac{4 t^{\frac{3}{2}}}{3} - (\frac{8 t^{\frac{7}{2}}}{21} + \frac{2 (2 t^{\frac{5}{2}})}{5}) + C$

Step 11.3.6

Multiply $2$ by $2$ .

$\frac{2 t^{\frac{7}{2}}}{3} + 2 t^{\frac{5}{2}} + \frac{4 t^{\frac{3}{2}}}{3} - (\frac{8 t^{\frac{7}{2}}}{21} + \frac{4 t^{\frac{5}{2}}}{5}) + C$

Step 11.3.7

To write $- (\frac{8 t^{\frac{7}{2}}}{21} + \frac{4 t^{\frac{5}{2}}}{5})$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$2 t^{\frac{5}{2}} + \frac{4 t^{\frac{3}{2}}}{3} + \frac{2 t^{\frac{7}{2}}}{3} - (\frac{8 t^{\frac{7}{2}}}{21} + \frac{4 t^{\frac{5}{2}}}{5}) \cdot \frac{3}{3} + C$

Step 11.3.8

Combine $- (\frac{8 t^{\frac{7}{2}}}{21} + \frac{4 t^{\frac{5}{2}}}{5})$ and $\frac{3}{3}$ .

$2 t^{\frac{5}{2}} + \frac{4 t^{\frac{3}{2}}}{3} + \frac{2 t^{\frac{7}{2}}}{3} + \frac{- (\frac{8 t^{\frac{7}{2}}}{21} + \frac{4 t^{\frac{5}{2}}}{5}) \cdot 3}{3} + C$

Step 11.3.9

Combine the numerators over the common denominator.

$2 t^{\frac{5}{2}} + \frac{4 t^{\frac{3}{2}}}{3} + \frac{2 t^{\frac{7}{2}} - (\frac{8 t^{\frac{7}{2}}}{21} + \frac{4 t^{\frac{5}{2}}}{5}) \cdot 3}{3} + C$

Step 11.3.10

Multiply $3$ by $- 1$ .

$2 t^{\frac{5}{2}} + \frac{4 t^{\frac{3}{2}}}{3} + \frac{2 t^{\frac{7}{2}} - 3 (\frac{8 t^{\frac{7}{2}}}{21} + \frac{4 t^{\frac{5}{2}}}{5})}{3} + C$

Step 11.4

Reorder terms.

$2 t^{\frac{5}{2}} + \frac{4}{3} t^{\frac{3}{2}} + \frac{1}{3} (2 t^{\frac{7}{2}} - 3 (\frac{8}{21} t^{\frac{7}{2}} + \frac{4}{5} t^{\frac{5}{2}})) + C$