Calculus Examples

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

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{t}$ as $t^{\frac{1}{3}}$ .

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

Step 2

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

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

Step 3

Simplify.

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

Reorder $t^{2}$ and $- 1$ .

$\int t^{2} t^{\frac{1}{3}} + t^{2} t^{\frac{1}{2}} - 1 \cdot t^{2} t^{- \frac{1}{2}} d t$

Step 3.4

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

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

Step 3.5

To write $2$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\int t^{2 \cdot \frac{3}{3} + \frac{1}{3}} + t^{2} t^{\frac{1}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.6

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

$\int t^{\frac{2 \cdot 3}{3} + \frac{1}{3}} + t^{2} t^{\frac{1}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.7

Combine the numerators over the common denominator.

$\int t^{\frac{2 \cdot 3 + 1}{3}} + t^{2} t^{\frac{1}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.8

Simplify the numerator.

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Step 3.8.1

Multiply $2$ by $3$ .

$\int t^{\frac{6 + 1}{3}} + t^{2} t^{\frac{1}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.8.2

Add $6$ and $1$ .

$\int t^{\frac{7}{3}} + t^{2} t^{\frac{1}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.9

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

$\int t^{\frac{7}{3}} + t^{2 + \frac{1}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.10

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int t^{\frac{7}{3}} + t^{2 \cdot \frac{2}{2} + \frac{1}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.11

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

$\int t^{\frac{7}{3}} + t^{\frac{2 \cdot 2}{2} + \frac{1}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.12

Combine the numerators over the common denominator.

$\int t^{\frac{7}{3}} + t^{\frac{2 \cdot 2 + 1}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.13

Simplify the numerator.

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Step 3.13.1

Multiply $2$ by $2$ .

$\int t^{\frac{7}{3}} + t^{\frac{4 + 1}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.13.2

Add $4$ and $1$ .

$\int t^{\frac{7}{3}} + t^{\frac{5}{2}} - 1 t^{2} t^{- \frac{1}{2}} d t$

Step 3.14

Factor out negative.

$\int t^{\frac{7}{3}} + t^{\frac{5}{2}} - (t^{2} t^{- \frac{1}{2}}) d t$

Step 3.15

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

$\int t^{\frac{7}{3}} + t^{\frac{5}{2}} - t^{2 - \frac{1}{2}} d t$

Step 3.16

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\int t^{\frac{7}{3}} + t^{\frac{5}{2}} - t^{2 \cdot \frac{2}{2} - \frac{1}{2}} d t$

Step 3.17

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

$\int t^{\frac{7}{3}} + t^{\frac{5}{2}} - t^{\frac{2 \cdot 2}{2} - \frac{1}{2}} d t$

Step 3.18

Combine the numerators over the common denominator.

$\int t^{\frac{7}{3}} + t^{\frac{5}{2}} - t^{\frac{2 \cdot 2 - 1}{2}} d t$

Step 3.19

Simplify the numerator.

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Step 3.19.1

Multiply $2$ by $2$ .

$\int t^{\frac{7}{3}} + t^{\frac{5}{2}} - t^{\frac{4 - 1}{2}} d t$

Step 3.19.2

Subtract $1$ from $4$ .

$\int t^{\frac{7}{3}} + t^{\frac{5}{2}} - t^{\frac{3}{2}} d t$

Step 4

Split the single integral into multiple integrals.

$\int t^{\frac{7}{3}} d t + \int t^{\frac{5}{2}} d t + \int - t^{\frac{3}{2}} d t$

Step 5

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

$\frac{3}{10} t^{\frac{10}{3}} + C + \int t^{\frac{5}{2}} d t + \int - t^{\frac{3}{2}} d t$

Step 6

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

$\frac{3}{10} t^{\frac{10}{3}} + C + \frac{2}{7} t^{\frac{7}{2}} + C + \int - t^{\frac{3}{2}} d t$

Step 7

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

$\frac{3}{10} t^{\frac{10}{3}} + C + \frac{2}{7} t^{\frac{7}{2}} + C - \int t^{\frac{3}{2}} d t$

Step 8

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

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

Step 9

Simplify.

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Step 9.1

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

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

Step 9.2

Simplify.

$\frac{3}{10} t^{\frac{10}{3}} + \frac{2}{7} t^{\frac{7}{2}} - \frac{2}{5} t^{\frac{5}{2}} + C$