Calculus Examples

$\int (2 x^{2} + \frac{3}{x} + \frac{5}{x^{7}} - 4 \sin (x) + 2 e^{x}) d x$

Step 1

Remove parentheses.

$\int 2 x^{2} + \frac{3}{x} + \frac{5}{x^{7}} - 4 \sin (x) + 2 e^{x} d x$

Step 2

Split the single integral into multiple integrals.

$\int 2 x^{2} d x + \int \frac{3}{x} d x + \int \frac{5}{x^{7}} d x + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 3

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

$2 \int x^{2} d x + \int \frac{3}{x} d x + \int \frac{5}{x^{7}} d x + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 4

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

$2 (\frac{1}{3} x^{3} + C) + \int \frac{3}{x} d x + \int \frac{5}{x^{7}} d x + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 5

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

$2 (\frac{1}{3} x^{3} + C) + 3 \int \frac{1}{x} d x + \int \frac{5}{x^{7}} d x + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 6

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

$2 (\frac{1}{3} x^{3} + C) + 3 (\ln (| x |) + C) + \int \frac{5}{x^{7}} d x + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 7

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

$2 (\frac{1}{3} x^{3} + C) + 3 (\ln (| x |) + C) + 5 \int \frac{1}{x^{7}} d x + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 8

Simplify the expression.

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Step 8.1

Move $x^{7}$ out of the denominator by raising it to the $- 1$ power.

$2 (\frac{1}{3} x^{3} + C) + 3 (\ln (| x |) + C) + 5 \int {(x^{7})}^{- 1} d x + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 8.2

Simplify.

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Step 8.2.1

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

$2 (\frac{x^{3}}{3} + C) + 3 (\ln (| x |) + C) + 5 \int {(x^{7})}^{- 1} d x + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 8.2.2

Multiply the exponents in ${(x^{7})}^{- 1}$ .

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Step 8.2.2.1

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

$2 (\frac{x^{3}}{3} + C) + 3 (\ln (| x |) + C) + 5 \int x^{7 \cdot - 1} d x + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 8.2.2.2

Multiply $7$ by $- 1$ .

$2 (\frac{x^{3}}{3} + C) + 3 (\ln (| x |) + C) + 5 \int x^{- 7} d x + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 9

By the Power Rule, the integral of $x^{- 7}$ with respect to $x$ is $- \frac{1}{6} x^{- 6}$ .

$2 (\frac{x^{3}}{3} + C) + 3 (\ln (| x |) + C) + 5 (- \frac{1}{6} x^{- 6} + C) + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 10

Simplify.

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Step 10.1

Combine $x^{- 6}$ and $\frac{1}{6}$ .

$2 (\frac{x^{3}}{3} + C) + 3 (\ln (| x |) + C) + 5 (- \frac{x^{- 6}}{6} + C) + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 10.2

Move $x^{- 6}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$2 (\frac{x^{3}}{3} + C) + 3 (\ln (| x |) + C) + 5 (- \frac{1}{6 x^{6}} + C) + \int - 4 \sin (x) d x + \int 2 e^{x} d x$

Step 11

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

$2 (\frac{x^{3}}{3} + C) + 3 (\ln (| x |) + C) + 5 (- \frac{1}{6 x^{6}} + C) - 4 \int \sin (x) d x + \int 2 e^{x} d x$

Step 12

The integral of $\sin (x)$ with respect to $x$ is $- \cos (x)$ .

$2 (\frac{x^{3}}{3} + C) + 3 (\ln (| x |) + C) + 5 (- \frac{1}{6 x^{6}} + C) - 4 (- \cos (x) + C) + \int 2 e^{x} d x$

Step 13

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

$2 (\frac{x^{3}}{3} + C) + 3 (\ln (| x |) + C) + 5 (- \frac{1}{6 x^{6}} + C) - 4 (- \cos (x) + C) + 2 \int e^{x} d x$

Step 14

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

$2 (\frac{x^{3}}{3} + C) + 3 (\ln (| x |) + C) + 5 (- \frac{1}{6 x^{6}} + C) - 4 (- \cos (x) + C) + 2 (e^{x} + C)$

Step 15

Simplify.

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Step 15.1

Simplify.

$\frac{2 x^{3}}{3} + 3 \ln (| x |) - \frac{5}{6 x^{6}} + 4 \cos (x) + 2 e^{x} + C$

Step 15.2

Reorder terms.

$\frac{2}{3} x^{3} + 3 \ln (| x |) - \frac{5}{6 x^{6}} + 4 \cos (x) + 2 e^{x} + C$