Calculus Examples

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

Step 1

Move $t^{\frac{4}{3}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2

Multiply the exponents in ${(t^{\frac{4}{3}})}^{- 1}$ .

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Step 2.1

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

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

Step 2.2

Multiply $\frac{4}{3} \cdot - 1$ .

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Step 2.2.1

Combine $\frac{4}{3}$ and $- 1$ .

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

Step 2.2.2

Multiply $4$ by $- 1$ .

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

Step 2.3

Move the negative in front of the fraction.

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

Step 3

Expand $(5 t^{2} + 7) t^{- \frac{4}{3}}$ .

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Step 3.1

Apply the distributive property.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 3.5

Combine the numerators over the common denominator.

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

Step 3.6

Simplify the numerator.

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Step 3.6.1

Multiply $2$ by $3$ .

$\int 5 t^{\frac{6 - 4}{3}} + 7 t^{- \frac{4}{3}} d t$

Step 3.6.2

Subtract $4$ from $6$ .

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

Step 3.7

Reorder $5 t^{\frac{2}{3}}$ and $7 t^{- \frac{4}{3}}$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

Since $7$ is constant with respect to $t$ , move $7$ out of the integral.

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

Step 6

By the Power Rule, the integral of $t^{- \frac{4}{3}}$ with respect to $t$ is $- 3 t^{- \frac{1}{3}}$ .

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

Step 7

Since $5$ is constant with respect to $t$ , move $5$ out of the integral.

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

Step 8

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

$7 (- 3 t^{- \frac{1}{3}} + C) + 5 (\frac{3}{5} t^{\frac{5}{3}} + C)$

Step 9

Simplify.

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Step 9.1

Simplify.

$7 (- 3 \frac{1}{t^{\frac{1}{3}}}) + 5 (\frac{3}{5} t^{\frac{5}{3}}) + C$

Step 9.2

Simplify.

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Step 9.2.1

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

$7 \frac{- 3}{t^{\frac{1}{3}}} + 5 (\frac{3}{5} t^{\frac{5}{3}}) + C$

Step 9.2.2

Move the negative in front of the fraction.

$7 (- \frac{3}{t^{\frac{1}{3}}}) + 5 (\frac{3}{5} t^{\frac{5}{3}}) + C$

Step 9.2.3

Multiply $- 1$ by $7$ .

$- 7 \frac{3}{t^{\frac{1}{3}}} + 5 (\frac{3}{5} t^{\frac{5}{3}}) + C$

Step 9.2.4

Combine $- 7$ and $\frac{3}{t^{\frac{1}{3}}}$ .

$\frac{- 7 \cdot 3}{t^{\frac{1}{3}}} + 5 (\frac{3}{5} t^{\frac{5}{3}}) + C$

Step 9.2.5

Multiply $- 7$ by $3$ .

$\frac{- 21}{t^{\frac{1}{3}}} + 5 (\frac{3}{5} t^{\frac{5}{3}}) + C$

Step 9.2.6

Move the negative in front of the fraction.

$- \frac{21}{t^{\frac{1}{3}}} + 5 (\frac{3}{5} t^{\frac{5}{3}}) + C$

Step 9.2.7

Combine $\frac{3}{5}$ and $t^{\frac{5}{3}}$ .

$- \frac{21}{t^{\frac{1}{3}}} + 5 \frac{3 t^{\frac{5}{3}}}{5} + C$

Step 9.2.8

Combine $5$ and $\frac{3 t^{\frac{5}{3}}}{5}$ .

$- \frac{21}{t^{\frac{1}{3}}} + \frac{5 (3 t^{\frac{5}{3}})}{5} + C$

Step 9.2.9

Multiply $3$ by $5$ .

$- \frac{21}{t^{\frac{1}{3}}} + \frac{15 t^{\frac{5}{3}}}{5} + C$

Step 9.2.10

Factor $5$ out of $15 t^{\frac{5}{3}}$ .

$- \frac{21}{t^{\frac{1}{3}}} + \frac{5 (3 t^{\frac{5}{3}})}{5} + C$

Step 9.2.11

Cancel the common factors.

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Step 9.2.11.1

Factor $5$ out of $5$ .

$- \frac{21}{t^{\frac{1}{3}}} + \frac{5 (3 t^{\frac{5}{3}})}{5 (1)} + C$

Step 9.2.11.2

Cancel the common factor.

$- \frac{21}{t^{\frac{1}{3}}} + \frac{5 (3 t^{\frac{5}{3}})}{5 \cdot 1} + C$

Step 9.2.11.3

Rewrite the expression.

$- \frac{21}{t^{\frac{1}{3}}} + \frac{3 t^{\frac{5}{3}}}{1} + C$

Step 9.2.11.4

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

$- \frac{21}{t^{\frac{1}{3}}} + 3 t^{\frac{5}{3}} + C$