Calculus Examples

$f (t) = \frac{3}{\sqrt{t}}$

Step 1

The function $F (t)$ can be found by finding the indefinite integral of the derivative $f (t)$ .

$F (t) = \int f (t) d t$

Step 2

Set up the integral to solve.

$F (t) = \int \frac{3}{\sqrt{t}} d t$

Step 3

Since $3$ is constant with respect to $t$ , move $3$ out of the integral.

$3 \int \frac{1}{\sqrt{t}} d t$

Step 4

Apply basic rules of exponents.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

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

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Step 4.3.1

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

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

Step 4.3.2

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

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

Step 4.3.3

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

Simplify the answer.

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Step 6.1

Rewrite $3 (2 t^{\frac{1}{2}} + C)$ as $3 \cdot 2 t^{\frac{1}{2}} + C$ .

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

Step 6.2

Multiply $3$ by $2$ .

$6 t^{\frac{1}{2}} + C$

Step 7

The answer is the antiderivative of the function $f (t) = \frac{3}{\sqrt{t}}$ .

$F (t) =$ $6 t^{\frac{1}{2}} + C$