Calculus Examples

$\int \frac{x^{4} - 3 x^{2} + 5}{x^{4}} d x$

Step 1

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

$\int (x^{4} - 3 x^{2} + 5) {(x^{4})}^{- 1} d x$

Step 2

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

Tap for more steps...

Step 2.1

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

$\int (x^{4} - 3 x^{2} + 5) x^{4 \cdot - 1} d x$

Step 2.2

Multiply $4$ by $- 1$ .

$\int (x^{4} - 3 x^{2} + 5) x^{- 4} d x$

Step 3

Expand $(x^{4} - 3 x^{2} + 5) x^{- 4}$ .

Tap for more steps...

Step 3.1

Apply the distributive property.

$\int (x^{4} - 3 x^{2}) x^{- 4} + 5 x^{- 4} d x$

Step 3.2

Apply the distributive property.

$\int x^{4} x^{- 4} - 3 x^{2} x^{- 4} + 5 x^{- 4} d x$

Step 3.3

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

$\int x^{4 - 4} - 3 x^{2} x^{- 4} + 5 x^{- 4} d x$

Step 3.4

Subtract $4$ from $4$ .

$\int x^{0} - 3 x^{2} x^{- 4} + 5 x^{- 4} d x$

Step 3.5

Anything raised to $0$ is $1$ .

$\int 1 - 3 x^{2} x^{- 4} + 5 x^{- 4} d x$

Step 3.6

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

$\int 1 - 3 x^{2 - 4} + 5 x^{- 4} d x$

Step 3.7

Subtract $4$ from $2$ .

$\int 1 - 3 x^{- 2} + 5 x^{- 4} d x$

Step 3.8

Reorder $1$ and $- 3 x^{- 2}$ .

$\int - 3 x^{- 2} + 1 + 5 x^{- 4} d x$

Step 3.9

Move $1$ .

$\int - 3 x^{- 2} + 5 x^{- 4} + 1 d x$

Step 4

Split the single integral into multiple integrals.

$\int - 3 x^{- 2} d x + \int 5 x^{- 4} d x + \int d x$

Step 5

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

$- 3 \int x^{- 2} d x + \int 5 x^{- 4} d x + \int d x$

Step 6

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

$- 3 (- x^{- 1} + C) + \int 5 x^{- 4} d x + \int d x$

Step 7

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$- 3 (- x^{- 1} + C) + 5 \int x^{- 4} d x + \int d x$

Step 8

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

$- 3 (- x^{- 1} + C) + 5 (- \frac{1}{3} x^{- 3} + C) + \int d x$

Step 9

Simplify.

Tap for more steps...

Step 9.1

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

$- 3 (- x^{- 1} + C) + 5 (- \frac{x^{- 3}}{3} + C) + \int d x$

Step 9.2

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

$- 3 (- x^{- 1} + C) + 5 (- \frac{1}{3 x^{3}} + C) + \int d x$

Step 10

Apply the constant rule.

$- 3 (- x^{- 1} + C) + 5 (- \frac{1}{3 x^{3}} + C) + x + C$

Step 11

Simplify.

$\frac{3}{x} - \frac{5}{3 x^{3}} + x + C$