Calculus Examples

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

Step 1

Simplify the expression.

Tap for more steps...

Step 1.1

Simplify.

Tap for more steps...

Step 1.1.1

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

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

Step 1.1.2

Multiply $t$ by $t^{\frac{1}{2}}$ by adding the exponents.

Tap for more steps...

Step 1.1.2.1

Multiply $t$ by $t^{\frac{1}{2}}$ .

Tap for more steps...

Step 1.1.2.1.1

Raise $t$ to the power of $1$ .

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

Step 1.1.2.1.2

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

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

Step 1.1.2.2

Write $1$ as a fraction with a common denominator.

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

Step 1.1.2.3

Combine the numerators over the common denominator.

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

Step 1.1.2.4

Add $2$ and $1$ .

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

Step 1.2

Apply basic rules of exponents.

Tap for more steps...

Step 1.2.1

Move $t^{\frac{3}{2}}$ out of the denominator by raising it to the $- 1$ power.

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

Step 1.2.2

Multiply the exponents in ${(t^{\frac{3}{2}})}^{- 1}$ .

Tap for more steps...

Step 1.2.2.1

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

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

Step 1.2.2.2

Multiply $\frac{3}{2} \cdot - 1$ .

Tap for more steps...

Step 1.2.2.2.1

Combine $\frac{3}{2}$ and $- 1$ .

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

Step 1.2.2.2.2

Multiply $3$ by $- 1$ .

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

Step 1.2.2.3

Move the negative in front of the fraction.

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

Step 2

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

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

Step 3

Simplify the answer.

Tap for more steps...

Step 3.1

Rewrite $- 2 t^{- \frac{1}{2}} + C$ as $- 2 \frac{1}{t^{\frac{1}{2}}} + C$ .

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

Step 3.2

Simplify.

Tap for more steps...

Step 3.2.1

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

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

Step 3.2.2

Move the negative in front of the fraction.

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