Calculus Examples

$\int \frac{27 t^{3} - 1}{\sqrt[3]{t}} d t$

Step 1

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

$\int \frac{27 t^{3} - 1}{t^{\frac{1}{3}}} d t$

Step 2

Move $t^{\frac{1}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\int (27 t^{3} - 1) {(t^{\frac{1}{3}})}^{- 1} d t$

Step 3

Multiply the exponents in ${(t^{\frac{1}{3}})}^{- 1}$ .

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Step 3.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\int (27 t^{3} - 1) t^{\frac{1}{3} \cdot - 1} d t$

Step 3.2

Combine $\frac{1}{3}$ and $- 1$ .

$\int (27 t^{3} - 1) t^{\frac{- 1}{3}} d t$

Step 3.3

Move the negative in front of the fraction.

$\int (27 t^{3} - 1) t^{- \frac{1}{3}} d t$

Step 4

Expand $(27 t^{3} - 1) t^{- \frac{1}{3}}$ .

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Step 4.1

Apply the distributive property.

$\int 27 t^{3} t^{- \frac{1}{3}} - 1 t^{- \frac{1}{3}} d t$

Step 4.2

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

$\int 27 t^{3 - \frac{1}{3}} - 1 t^{- \frac{1}{3}} d t$

Step 4.3

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

$\int 27 t^{3 \cdot \frac{3}{3} - \frac{1}{3}} - 1 t^{- \frac{1}{3}} d t$

Step 4.4

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

$\int 27 t^{\frac{3 \cdot 3}{3} - \frac{1}{3}} - 1 t^{- \frac{1}{3}} d t$

Step 4.5

Combine the numerators over the common denominator.

$\int 27 t^{\frac{3 \cdot 3 - 1}{3}} - 1 t^{- \frac{1}{3}} d t$

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $3$ by $3$ .

$\int 27 t^{\frac{9 - 1}{3}} - 1 t^{- \frac{1}{3}} d t$

Step 4.6.2

Subtract $1$ from $9$ .

$\int 27 t^{\frac{8}{3}} - 1 t^{- \frac{1}{3}} d t$

Step 5

Split the single integral into multiple integrals.

$\int 27 t^{\frac{8}{3}} d t + \int - 1 t^{- \frac{1}{3}} d t$

Step 6

Since $27$ is constant with respect to $t$ , move $27$ out of the integral.

$27 \int t^{\frac{8}{3}} d t + \int - 1 t^{- \frac{1}{3}} d t$

Step 7

By the Power Rule, the integral of $t^{\frac{8}{3}}$ with respect to $t$ is $\frac{3}{11} t^{\frac{11}{3}}$ .

$27 (\frac{3}{11} t^{\frac{11}{3}} + C) + \int - 1 t^{- \frac{1}{3}} d t$

Step 8

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

$27 (\frac{3}{11} t^{\frac{11}{3}} + C) - \int t^{- \frac{1}{3}} d t$

Step 9

By the Power Rule, the integral of $t^{- \frac{1}{3}}$ with respect to $t$ is $\frac{3}{2} t^{\frac{2}{3}}$ .

$27 (\frac{3}{11} t^{\frac{11}{3}} + C) - (\frac{3}{2} t^{\frac{2}{3}} + C)$

Step 10

Simplify.

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Step 10.1

Simplify.

$27 (\frac{3}{11}) t^{\frac{11}{3}} - \frac{3}{2} t^{\frac{2}{3}} + C$

Step 10.2

Simplify.

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Step 10.2.1

Combine $27$ and $\frac{3}{11}$ .

$\frac{27 \cdot 3}{11} t^{\frac{11}{3}} - \frac{3}{2} t^{\frac{2}{3}} + C$

Step 10.2.2

Multiply $27$ by $3$ .

$\frac{81}{11} t^{\frac{11}{3}} - \frac{3}{2} t^{\frac{2}{3}} + C$