Calculus Examples

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

Step 1

Simplify.

Tap for more steps...

Step 1.1

Multiply $\frac{1}{t}$ by $\frac{2}{t^{2}} - \frac{3}{t^{3}}$ .

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

Step 1.2

Simplify the numerator.

Tap for more steps...

Step 1.2.1

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

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

Step 1.2.2

Write each expression with a common denominator of $t^{3}$ , by multiplying each by an appropriate factor of $1$ .

Tap for more steps...

Step 1.2.2.1

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

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

Step 1.2.2.2

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

Tap for more steps...

Step 1.2.2.2.1

Multiply $t^{2}$ by $t$ .

Tap for more steps...

Step 1.2.2.2.1.1

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

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

Step 1.2.2.2.1.2

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

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

Step 1.2.2.2.2

Add $2$ and $1$ .

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

Step 1.2.3

Combine the numerators over the common denominator.

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

Step 1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.4

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

Tap for more steps...

Step 1.4.1

Multiply $\frac{2 t - 3}{t^{3}}$ by $\frac{1}{t}$ .

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

Step 1.4.2

Multiply $t^{3}$ by $t$ by adding the exponents.

Tap for more steps...

Step 1.4.2.1

Multiply $t^{3}$ by $t$ .

Tap for more steps...

Step 1.4.2.1.1

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

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

Step 1.4.2.1.2

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

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

Step 1.4.2.2

Add $3$ and $1$ .

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

Step 2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1

Move $t^{4}$ out of the denominator by raising it to the $- 1$ power.

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

Step 2.2

Multiply the exponents in ${(t^{4})}^{- 1}$ .

Tap for more steps...

Step 2.2.1

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

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

Step 2.2.2

Multiply $4$ by $- 1$ .

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

Step 3

Multiply $(2 t - 3) t^{- 4}$ .

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

Step 4

Multiply $t$ by $t^{- 4}$ by adding the exponents.

Tap for more steps...

Step 4.1

Move $t^{- 4}$ .

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

Step 4.2

Multiply $t^{- 4}$ by $t$ .

Tap for more steps...

Step 4.2.1

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

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

Step 4.2.2

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

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

Step 4.3

Add $- 4$ and $1$ .

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

Step 5

Split the single integral into multiple integrals.

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

Step 6

Since $2$ is constant with respect to $t$ , move $2$ out of the integral.

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

Step 7

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

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

Step 8

Simplify.

Tap for more steps...

Step 8.1

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

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

Step 8.2

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

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

Step 9

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

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

Step 10

By the Power Rule, the integral of $t^{- 4}$ with respect to $t$ is $- \frac{1}{3} t^{- 3}$ .

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

Step 11

Simplify.

Tap for more steps...

Step 11.1

Simplify.

Tap for more steps...

Step 11.1.1

Combine $t^{- 3}$ and $\frac{1}{3}$ .

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

Step 11.1.2

Move $t^{- 3}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 11.2

Simplify.

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

Step 11.3

Simplify.

Tap for more steps...

Step 11.3.1

Move the negative in front of the fraction.

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

Step 11.3.2

Multiply $- 1$ by $- 3$ .

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

Step 11.3.3

Combine $3$ and $\frac{1}{3 t^{3}}$ .

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

Step 11.3.4

Cancel the common factor of $3$ .

Tap for more steps...

Step 11.3.4.1

Cancel the common factor.

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

Step 11.3.4.2

Rewrite the expression.

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