Calculus Examples

$\int [3 \sec^{2} (t) + \frac{1}{\sqrt{t}}] d t$

Step 1

Remove parentheses.

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

Step 2

Split the single integral into multiple integrals.

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

Step 3

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

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

Step 4

Since the derivative of $\tan (t)$ is $\sec^{2} (t)$ , the integral of $\sec^{2} (t)$ is $\tan (t)$ .

$3 (\tan (t) + C) + \int \frac{1}{\sqrt{t}} d t$

Step 5

Apply basic rules of exponents.

Tap for more steps...

Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

Tap for more steps...

Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

Move the negative in front of the fraction.

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

Step 6

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

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

Step 7

Simplify.

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