Calculus Examples

$\frac{1}{\sqrt{t}}$

Step 1

Apply basic rules of exponents.

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Step 1.1

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

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

Step 1.2

Move $t^{\frac{1}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 1.3

Multiply the exponents in ${(t^{\frac{1}{2}})}^{- 1}$ .

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Step 1.3.1

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

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

Step 1.3.2

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

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

Step 1.3.3

Move the negative in front of the fraction.

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

Step 2

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

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