Calculus Examples

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

Step 1

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

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Multiply $4$ by $- 1$ .

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

Step 3

Expand $(x^{5} + 2 x^{2} - 1) x^{- 4}$ .

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Step 3.1

Apply the distributive property.

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

Step 3.2

Apply the distributive property.

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

Step 3.3

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

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

Step 3.4

Subtract $4$ from $5$ .

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

Step 3.5

Simplify.

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

Step 3.6

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

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

Step 3.7

Subtract $4$ from $2$ .

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

Step 4

Split the single integral into multiple integrals.

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

Step 5

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

$\frac{1}{2} x^{2} + C + \int 2 x^{- 2} d x + \int - 1 x^{- 4} d x$

Step 6

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

$\frac{1}{2} x^{2} + C + 2 \int x^{- 2} d x + \int - 1 x^{- 4} d x$

Step 7

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

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

Step 8

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

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

Step 9

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

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

Step 10

Simplify.

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Step 10.1

Simplify.

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Step 10.1.1

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

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

Step 10.1.2

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

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

Step 10.2

Simplify.

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

Step 10.3

Simplify.

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Step 10.3.1

Multiply $- 1$ by $- 1$ .

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

Step 10.3.2

Multiply $\frac{1}{3 x^{3}}$ by $1$ .

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

Step 10.4

Reorder terms.

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