Calculus Examples

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

Step 1

Simplify.

Tap for more steps...

Step 1.1

Simplify the numerator.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

Factor $t^{\frac{1}{3}}$ out of $t^{3} + t^{\frac{1}{3}}$ .

Tap for more steps...

Step 1.1.2.1

Factor $t^{\frac{1}{3}}$ out of $t^{3}$ .

$\int \frac{t^{\frac{1}{3}} t^{\frac{8}{3}} + t^{\frac{1}{3}}}{t^{2}} d t$

Step 1.1.2.2

Multiply by $1$ .

$\int \frac{t^{\frac{1}{3}} t^{\frac{8}{3}} + t^{\frac{1}{3}} \cdot 1}{t^{2}} d t$

Step 1.1.2.3

Factor $t^{\frac{1}{3}}$ out of $t^{\frac{1}{3}} t^{\frac{8}{3}} + t^{\frac{1}{3}} \cdot 1$ .

$\int \frac{t^{\frac{1}{3}} (t^{\frac{8}{3}} + 1)}{t^{2}} d t$

Step 1.2

Move $t^{\frac{1}{3}}$ to the denominator using the negative exponent rule $b^{n} = \frac{1}{b^{- n}}$ .

$\int \frac{t^{\frac{8}{3}} + 1}{t^{2} t^{- \frac{1}{3}}} d t$

Step 1.3

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

Tap for more steps...

Step 1.3.1

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

$\int \frac{t^{\frac{8}{3}} + 1}{t^{2 - \frac{1}{3}}} d t$

Step 1.3.2

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

$\int \frac{t^{\frac{8}{3}} + 1}{t^{2 \cdot \frac{3}{3} - \frac{1}{3}}} d t$

Step 1.3.3

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

$\int \frac{t^{\frac{8}{3}} + 1}{t^{\frac{2 \cdot 3}{3} - \frac{1}{3}}} d t$

Step 1.3.4

Combine the numerators over the common denominator.

$\int \frac{t^{\frac{8}{3}} + 1}{t^{\frac{2 \cdot 3 - 1}{3}}} d t$

Step 1.3.5

Simplify the numerator.

Tap for more steps...

Step 1.3.5.1

Multiply $2$ by $3$ .

$\int \frac{t^{\frac{8}{3}} + 1}{t^{\frac{6 - 1}{3}}} d t$

Step 1.3.5.2

Subtract $1$ from $6$ .

$\int \frac{t^{\frac{8}{3}} + 1}{t^{\frac{5}{3}}} d t$

Step 2

Move $t^{\frac{5}{3}}$ out of the denominator by raising it to the $- 1$ power.

$\int (t^{\frac{8}{3}} + 1) {(t^{\frac{5}{3}})}^{- 1} d t$

Step 3

Multiply the exponents in ${(t^{\frac{5}{3}})}^{- 1}$ .

Tap for more steps...

Step 3.1

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

$\int (t^{\frac{8}{3}} + 1) t^{\frac{5}{3} \cdot - 1} d t$

Step 3.2

Multiply $\frac{5}{3} \cdot - 1$ .

Tap for more steps...

Step 3.2.1

Combine $\frac{5}{3}$ and $- 1$ .

$\int (t^{\frac{8}{3}} + 1) t^{\frac{5 \cdot - 1}{3}} d t$

Step 3.2.2

Multiply $5$ by $- 1$ .

$\int (t^{\frac{8}{3}} + 1) t^{\frac{- 5}{3}} d t$

Step 3.3

Move the negative in front of the fraction.

$\int (t^{\frac{8}{3}} + 1) t^{- \frac{5}{3}} d t$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Apply the distributive property.

$\int t^{\frac{8}{3}} t^{- \frac{5}{3}} + 1 t^{- \frac{5}{3}} d t$

Step 4.2

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

$\int t^{\frac{8}{3} - \frac{5}{3}} + 1 t^{- \frac{5}{3}} d t$

Step 4.3

Combine the numerators over the common denominator.

$\int t^{\frac{8 - 5}{3}} + 1 t^{- \frac{5}{3}} d t$

Step 4.4

Subtract $5$ from $8$ .

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

Step 4.5

Cancel the common factor of $3$ .

Tap for more steps...

Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

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

Step 4.6

Simplify.

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

Step 4.7

Multiply $t^{- \frac{5}{3}}$ by $1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify.

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