Calculus Examples

$\int 6 t^{2} \sqrt[3]{t} d t$

Step 1

Simplify.

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Step 1.1

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

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

Step 1.2

Multiply $t^{2}$ by $t^{\frac{1}{3}}$ by adding the exponents.

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Step 1.2.1

Move $t^{\frac{1}{3}}$ .

$\int 6 (t^{\frac{1}{3}} t^{2}) d t$

Step 1.2.2

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

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

Step 1.2.3

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

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

Step 1.2.4

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

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

Step 1.2.5

Combine the numerators over the common denominator.

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

Step 1.2.6

Simplify the numerator.

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Step 1.2.6.1

Multiply $2$ by $3$ .

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

Step 1.2.6.2

Add $1$ and $6$ .

$\int 6 t^{\frac{7}{3}} d t$

Step 2

Since $6$ is constant with respect to $t$ , move $6$ out of the integral.

$6 \int t^{\frac{7}{3}} d t$

Step 3

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

$6 (\frac{3}{10} t^{\frac{10}{3}} + C)$

Step 4

Simplify the answer.

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Step 4.1

Rewrite $6 (\frac{3}{10} t^{\frac{10}{3}} + C)$ as $6 (\frac{3}{10}) t^{\frac{10}{3}} + C$ .

$6 (\frac{3}{10}) t^{\frac{10}{3}} + C$

Step 4.2

Simplify.

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Step 4.2.1

Combine $6$ and $\frac{3}{10}$ .

$\frac{6 \cdot 3}{10} t^{\frac{10}{3}} + C$

Step 4.2.2

Multiply $6$ by $3$ .

$\frac{18}{10} t^{\frac{10}{3}} + C$

Step 4.2.3

Cancel the common factor of $18$ and $10$ .

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Step 4.2.3.1

Factor $2$ out of $18$ .

$\frac{2 (9)}{10} t^{\frac{10}{3}} + C$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $2$ out of $10$ .

$\frac{2 \cdot 9}{2 \cdot 5} t^{\frac{10}{3}} + C$

Step 4.2.3.2.2

Cancel the common factor.

$\frac{2 \cdot 9}{2 \cdot 5} t^{\frac{10}{3}} + C$

Step 4.2.3.2.3

Rewrite the expression.

$\frac{9}{5} t^{\frac{10}{3}} + C$