Calculus Examples

$\int \sqrt[3]{t} (5 t^{2} - 3 t + 2) d t$

Step 1

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = 5 t^{2} - 3 t + 2$ and $d v = \sqrt[3]{t}$ .

$(5 t^{2} - 3 t + 2) (\frac{3}{4} t^{\frac{4}{3}}) - \int \frac{3}{4} t^{\frac{4}{3}} (10 t - 3) d t$

Step 2

Simplify.

Tap for more steps...

Step 2.1

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

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{3}{4} t^{\frac{4}{3}} (10 t - 3) d t$

Step 2.2

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

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{3 t^{\frac{4}{3}}}{4} (10 t - 3) d t$

Step 3

Simplify.

Tap for more steps...

Step 3.1

Apply the distributive property.

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{3 t^{\frac{4}{3}}}{4} (10 t) + \frac{3 t^{\frac{4}{3}}}{4} \cdot - 3 d t$

Step 3.2

Reorder $\frac{3 t^{\frac{4}{3}}}{4}$ and $10$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int 10 \cdot \frac{3 t^{\frac{4}{3}}}{4} t + \frac{3 t^{\frac{4}{3}}}{4} \cdot - 3 d t$

Step 3.3

Reorder $\frac{3 t^{\frac{4}{3}}}{4}$ and $- 3$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int 10 \cdot \frac{3 t^{\frac{4}{3}}}{4} t - 3 \cdot \frac{3 t^{\frac{4}{3}}}{4} d t$

Step 3.4

Combine $10$ and $\frac{3 t^{\frac{4}{3}}}{4}$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{10 (3 t^{\frac{4}{3}})}{4} t - 3 \frac{3 t^{\frac{4}{3}}}{4} d t$

Step 3.5

Multiply $10$ by $3$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{30 t^{\frac{4}{3}}}{4} t - 3 \frac{3 t^{\frac{4}{3}}}{4} d t$

Step 3.6

Combine $\frac{30 t^{\frac{4}{3}}}{4}$ and $t$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{30 t^{\frac{4}{3}} t}{4} - 3 \frac{3 t^{\frac{4}{3}}}{4} d t$

Step 3.7

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

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{30 (t^{\frac{4}{3}} t^{1})}{4} - 3 \frac{3 t^{\frac{4}{3}}}{4} d t$

Step 3.8

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

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{30 t^{\frac{4}{3} + 1}}{4} - 3 \frac{3 t^{\frac{4}{3}}}{4} d t$

Step 3.9

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

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{30 t^{\frac{4}{3} + \frac{3}{3}}}{4} - 3 \frac{3 t^{\frac{4}{3}}}{4} d t$

Step 3.10

Combine the numerators over the common denominator.

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{30 t^{\frac{4 + 3}{3}}}{4} - 3 \frac{3 t^{\frac{4}{3}}}{4} d t$

Step 3.11

Add $4$ and $3$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{30 t^{\frac{7}{3}}}{4} - 3 \frac{3 t^{\frac{4}{3}}}{4} d t$

Step 3.12

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

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{30 t^{\frac{7}{3}}}{4} + \frac{- 3 (3 t^{\frac{4}{3}})}{4} d t$

Step 3.13

Multiply $- 3$ by $3$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{30 t^{\frac{7}{3}}}{4} + \frac{- 9 t^{\frac{4}{3}}}{4} d t$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Factor $2$ out of $30 t^{\frac{7}{3}}$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{2 (15 t^{\frac{7}{3}})}{4} + \frac{- 9 t^{\frac{4}{3}}}{4} d t$

Step 4.2

Cancel the common factors.

Tap for more steps...

Step 4.2.1

Factor $2$ out of $4$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{2 (15 t^{\frac{7}{3}})}{2 (2)} + \frac{- 9 t^{\frac{4}{3}}}{4} d t$

Step 4.2.2

Cancel the common factor.

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{2 (15 t^{\frac{7}{3}})}{2 \cdot 2} + \frac{- 9 t^{\frac{4}{3}}}{4} d t$

Step 4.2.3

Rewrite the expression.

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{15 t^{\frac{7}{3}}}{2} + \frac{- 9 t^{\frac{4}{3}}}{4} d t$

Step 4.3

Move the negative in front of the fraction.

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - \int \frac{15 t^{\frac{7}{3}}}{2} - \frac{9 t^{\frac{4}{3}}}{4} d t$

Step 5

Split the single integral into multiple integrals.

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - (\int \frac{15 t^{\frac{7}{3}}}{2} d t + \int - \frac{9 t^{\frac{4}{3}}}{4} d t)$

Step 6

Since $\frac{15}{2}$ is constant with respect to $t$ , move $\frac{15}{2}$ out of the integral.

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - (\frac{15}{2} \int t^{\frac{7}{3}} d t + \int - \frac{9 t^{\frac{4}{3}}}{4} d t)$

Step 7

By the Power Rule, the integral of $t^{\frac{7}{3}}$ with respect to $t$ is $\frac{3}{10} t^{\frac{10}{3}}$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - (\frac{15}{2} (\frac{3}{10} t^{\frac{10}{3}} + C) + \int - \frac{9 t^{\frac{4}{3}}}{4} d t)$

Step 8

Combine $\frac{3}{10}$ and $t^{\frac{10}{3}}$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - (\frac{15}{2} (\frac{3 t^{\frac{10}{3}}}{10} + C) + \int - \frac{9 t^{\frac{4}{3}}}{4} d t)$

Step 9

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

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - (\frac{15}{2} (\frac{3 t^{\frac{10}{3}}}{10} + C) - \int \frac{9 t^{\frac{4}{3}}}{4} d t)$

Step 10

Since $\frac{9}{4}$ is constant with respect to $t$ , move $\frac{9}{4}$ out of the integral.

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - (\frac{15}{2} (\frac{3 t^{\frac{10}{3}}}{10} + C) - (\frac{9}{4} \int t^{\frac{4}{3}} d t))$

Step 11

By the Power Rule, the integral of $t^{\frac{4}{3}}$ with respect to $t$ is $\frac{3}{7} t^{\frac{7}{3}}$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - (\frac{15}{2} (\frac{3 t^{\frac{10}{3}}}{10} + C) - \frac{9}{4} (\frac{3}{7} t^{\frac{7}{3}} + C))$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Combine $\frac{3}{7}$ and $t^{\frac{7}{3}}$ .

$(5 t^{2} - 3 t + 2) \frac{3 t^{\frac{4}{3}}}{4} - (\frac{15}{2} (\frac{3 t^{\frac{10}{3}}}{10} + C) - \frac{9}{4} (\frac{3 t^{\frac{7}{3}}}{7} + C))$

Step 12.2

Simplify.

$\frac{t^{\frac{10}{3}} \cdot 15}{4} + \frac{- 9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{15}{2} \cdot \frac{3 t^{\frac{10}{3}}}{10} - \frac{9}{4} \cdot \frac{3 t^{\frac{7}{3}}}{7}) + C$

Step 12.3

Simplify.

Tap for more steps...

Step 12.3.1

Move $15$ to the left of $t^{\frac{10}{3}}$ .

$\frac{15 \cdot t^{\frac{10}{3}}}{4} + \frac{- 9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{15}{2} \cdot \frac{3 t^{\frac{10}{3}}}{10} - \frac{9}{4} \cdot \frac{3 t^{\frac{7}{3}}}{7}) + C$

Step 12.3.2

Move the negative in front of the fraction.

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{15}{2} \cdot \frac{3 t^{\frac{10}{3}}}{10} - \frac{9}{4} \cdot \frac{3 t^{\frac{7}{3}}}{7}) + C$

Step 12.3.3

Multiply $\frac{15}{2}$ by $\frac{3 t^{\frac{10}{3}}}{10}$ .

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{15 (3 t^{\frac{10}{3}})}{2 \cdot 10} - \frac{9}{4} \cdot \frac{3 t^{\frac{7}{3}}}{7}) + C$

Step 12.3.4

Multiply $3$ by $15$ .

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{45 t^{\frac{10}{3}}}{2 \cdot 10} - \frac{9}{4} \cdot \frac{3 t^{\frac{7}{3}}}{7}) + C$

Step 12.3.5

Multiply $2$ by $10$ .

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{45 t^{\frac{10}{3}}}{20} - \frac{9}{4} \cdot \frac{3 t^{\frac{7}{3}}}{7}) + C$

Step 12.3.6

Factor $5$ out of $45 t^{\frac{10}{3}}$ .

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{5 (9 t^{\frac{10}{3}})}{20} - \frac{9}{4} \cdot \frac{3 t^{\frac{7}{3}}}{7}) + C$

Step 12.3.7

Cancel the common factors.

Tap for more steps...

Step 12.3.7.1

Factor $5$ out of $20$ .

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{5 (9 t^{\frac{10}{3}})}{5 (4)} - \frac{9}{4} \cdot \frac{3 t^{\frac{7}{3}}}{7}) + C$

Step 12.3.7.2

Cancel the common factor.

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{5 (9 t^{\frac{10}{3}})}{5 \cdot 4} - \frac{9}{4} \cdot \frac{3 t^{\frac{7}{3}}}{7}) + C$

Step 12.3.7.3

Rewrite the expression.

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{9 t^{\frac{10}{3}}}{4} - \frac{9}{4} \cdot \frac{3 t^{\frac{7}{3}}}{7}) + C$

Step 12.3.8

Multiply $\frac{3 t^{\frac{7}{3}}}{7}$ by $\frac{9}{4}$ .

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{9 t^{\frac{10}{3}}}{4} - \frac{3 t^{\frac{7}{3}} \cdot 9}{7 \cdot 4}) + C$

Step 12.3.9

Multiply $9$ by $3$ .

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{9 t^{\frac{10}{3}}}{4} - \frac{27 t^{\frac{7}{3}}}{7 \cdot 4}) + C$

Step 12.3.10

Multiply $7$ by $4$ .

$\frac{15 t^{\frac{10}{3}}}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} - (\frac{9 t^{\frac{10}{3}}}{4} - \frac{27 t^{\frac{7}{3}}}{28}) + C$

Step 12.3.11

To write $- (\frac{9 t^{\frac{10}{3}}}{4} - \frac{27 t^{\frac{7}{3}}}{28})$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$\frac{15 t^{\frac{10}{3}}}{4} - (\frac{9 t^{\frac{10}{3}}}{4} - \frac{27 t^{\frac{7}{3}}}{28}) \cdot \frac{4}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} + C$

Step 12.3.12

Combine $- (\frac{9 t^{\frac{10}{3}}}{4} - \frac{27 t^{\frac{7}{3}}}{28})$ and $\frac{4}{4}$ .

$\frac{15 t^{\frac{10}{3}}}{4} + \frac{- (\frac{9 t^{\frac{10}{3}}}{4} - \frac{27 t^{\frac{7}{3}}}{28}) \cdot 4}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} + C$

Step 12.3.13

Combine the numerators over the common denominator.

$\frac{15 t^{\frac{10}{3}} - (\frac{9 t^{\frac{10}{3}}}{4} - \frac{27 t^{\frac{7}{3}}}{28}) \cdot 4}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} + C$

Step 12.3.14

Multiply $4$ by $- 1$ .

$\frac{15 t^{\frac{10}{3}} - 4 (\frac{9 t^{\frac{10}{3}}}{4} - \frac{27 t^{\frac{7}{3}}}{28})}{4} - \frac{9 t^{\frac{7}{3}}}{4} + \frac{3 t^{\frac{4}{3}}}{2} + C$

Step 13

Reorder terms.

$\frac{1}{4} (15 t^{\frac{10}{3}} - 4 (\frac{9}{4} t^{\frac{10}{3}} - \frac{27}{28} t^{\frac{7}{3}})) - \frac{9}{4} t^{\frac{7}{3}} + \frac{3}{2} t^{\frac{4}{3}} + C$