Calculus Examples

$\int \frac{4}{x^{3}} - \frac{1}{x^{2}} + \frac{1}{x} - \sqrt{x} + 8 x^{3} - e^{x} d x$

Step 1

Split the single integral into multiple integrals.

$\int \frac{4}{x^{3}} d x + \int - \frac{1}{x^{2}} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 2

Since $4$ is constant with respect to $x$ , move $4$ out of the integral.

$4 \int \frac{1}{x^{3}} d x + \int - \frac{1}{x^{2}} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 3

Apply basic rules of exponents.

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Step 3.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$4 \int {(x^{3})}^{- 1} d x + \int - \frac{1}{x^{2}} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 3.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

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Step 3.2.1

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

$4 \int x^{3 \cdot - 1} d x + \int - \frac{1}{x^{2}} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 3.2.2

Multiply $3$ by $- 1$ .

$4 \int x^{- 3} d x + \int - \frac{1}{x^{2}} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 4

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

$4 (- \frac{1}{2} x^{- 2} + C) + \int - \frac{1}{x^{2}} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 5

Simplify.

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Step 5.1

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

$4 (- \frac{x^{- 2}}{2} + C) + \int - \frac{1}{x^{2}} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 5.2

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

$4 (- \frac{1}{2 x^{2}} + C) + \int - \frac{1}{x^{2}} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 6

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

$4 (- \frac{1}{2 x^{2}} + C) - \int \frac{1}{x^{2}} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 7

Apply basic rules of exponents.

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Step 7.1

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

$4 (- \frac{1}{2 x^{2}} + C) - \int {(x^{2})}^{- 1} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 7.2

Multiply the exponents in ${(x^{2})}^{- 1}$ .

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Step 7.2.1

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

$4 (- \frac{1}{2 x^{2}} + C) - \int x^{2 \cdot - 1} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 7.2.2

Multiply $2$ by $- 1$ .

$4 (- \frac{1}{2 x^{2}} + C) - \int x^{- 2} d x + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 8

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

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \int \frac{1}{x} d x + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 9

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \ln (| x |) + C + \int - \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 10

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

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \ln (| x |) + C - \int \sqrt{x} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 11

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

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \ln (| x |) + C - \int x^{\frac{1}{2}} d x + \int 8 x^{3} d x + \int - e^{x} d x$

Step 12

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

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \ln (| x |) + C - (\frac{2}{3} x^{\frac{3}{2}} + C) + \int 8 x^{3} d x + \int - e^{x} d x$

Step 13

Combine $\frac{2}{3}$ and $x^{\frac{3}{2}}$ .

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \ln (| x |) + C - (\frac{2 x^{\frac{3}{2}}}{3} + C) + \int 8 x^{3} d x + \int - e^{x} d x$

Step 14

Since $8$ is constant with respect to $x$ , move $8$ out of the integral.

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \ln (| x |) + C - (\frac{2 x^{\frac{3}{2}}}{3} + C) + 8 \int x^{3} d x + \int - e^{x} d x$

Step 15

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

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \ln (| x |) + C - (\frac{2 x^{\frac{3}{2}}}{3} + C) + 8 (\frac{1}{4} x^{4} + C) + \int - e^{x} d x$

Step 16

Combine $\frac{1}{4}$ and $x^{4}$ .

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \ln (| x |) + C - (\frac{2 x^{\frac{3}{2}}}{3} + C) + 8 (\frac{x^{4}}{4} + C) + \int - e^{x} d x$

Step 17

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

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \ln (| x |) + C - (\frac{2 x^{\frac{3}{2}}}{3} + C) + 8 (\frac{x^{4}}{4} + C) - \int e^{x} d x$

Step 18

The integral of $e^{x}$ with respect to $x$ is $e^{x}$ .

$4 (- \frac{1}{2 x^{2}} + C) - (- x^{- 1} + C) + \ln (| x |) + C - (\frac{2 x^{\frac{3}{2}}}{3} + C) + 8 (\frac{x^{4}}{4} + C) - (e^{x} + C)$

Step 19

Simplify.

$- \frac{2}{x^{2}} + \frac{1}{x} + \ln (| x |) - \frac{2 x^{\frac{3}{2}}}{3} + 2 x^{4} - e^{x} + C$

Step 20

Reorder terms.

$- \frac{2}{x^{2}} + \frac{1}{x} + \ln (| x |) - \frac{2}{3} x^{\frac{3}{2}} + 2 x^{4} - e^{x} + C$