Calculus Examples

$\int \frac{4 + \sqrt{t}}{t^{3}} d t$

Step 1

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

$\int (4 + \sqrt{t}) {(t^{3})}^{- 1} d t$

Step 2

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

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Step 2.1

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

$\int (4 + \sqrt{t}) t^{3 \cdot - 1} d t$

Step 2.2

Multiply $3$ by $- 1$ .

$\int (4 + \sqrt{t}) t^{- 3} d t$

Step 3

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

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

Step 4

Simplify.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

Combine the numerators over the common denominator.

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

Step 4.6

Simplify the numerator.

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Step 4.6.1

Multiply $- 3$ by $2$ .

$\int 4 t^{- 3} + t^{\frac{1 - 6}{2}} d t$

Step 4.6.2

Subtract $6$ from $1$ .

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

Step 5

Move the negative in front of the fraction.

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

Step 6

Split the single integral into multiple integrals.

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

Step 7

Since $4$ is constant with respect to $t$ , move $4$ out of the integral.

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

Step 8

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

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

Step 9

Simplify.

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Step 9.1

Combine $t^{- 2}$ and $\frac{1}{2}$ .

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

Step 9.2

Move $t^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 10

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

$4 (- \frac{1}{2 t^{2}} + C) - \frac{2}{3} t^{- \frac{3}{2}} + C$

Step 11

Simplify.

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Step 11.1

Simplify.

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

Step 11.2

Move the negative in front of the fraction.

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