Calculus Examples

$f (x) = 2 x^{\frac{2}{9}} - \frac{2}{x^{4}} + 6 x + 2 \cos (x) + \ln (5)$

Step 1

The function $F (x)$ can be found by finding the indefinite integral of the derivative $f (x)$ .

$F (x) = \int f (x) d x$

Step 2

Set up the integral to solve.

$F (x) = \int 2 x^{\frac{2}{9}} - \frac{2}{x^{4}} + 6 x + 2 \cos (x) + \ln (5) d x$

Step 3

Split the single integral into multiple integrals.

$\int 2 x^{\frac{2}{9}} d x + \int - \frac{2}{x^{4}} d x + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 4

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

$2 \int x^{\frac{2}{9}} d x + \int - \frac{2}{x^{4}} d x + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 5

By the Power Rule, the integral of $x^{\frac{2}{9}}$ with respect to $x$ is $\frac{9}{11} x^{\frac{11}{9}}$ .

$2 (\frac{9}{11} x^{\frac{11}{9}} + C) + \int - \frac{2}{x^{4}} d x + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 6

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

$2 (\frac{9}{11} x^{\frac{11}{9}} + C) - \int \frac{2}{x^{4}} d x + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 7

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

$2 (\frac{9}{11} x^{\frac{11}{9}} + C) - (2 \int \frac{1}{x^{4}} d x) + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 8

Simplify the expression.

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Step 8.1

Simplify.

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Step 8.1.1

Combine $\frac{9}{11}$ and $x^{\frac{11}{9}}$ .

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - (2 \int \frac{1}{x^{4}} d x) + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 8.1.2

Multiply $2$ by $- 1$ .

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 \int \frac{1}{x^{4}} d x + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 8.2

Apply basic rules of exponents.

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Step 8.2.1

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

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 \int {(x^{4})}^{- 1} d x + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 8.2.2

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

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Step 8.2.2.1

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

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 \int x^{4 \cdot - 1} d x + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 8.2.2.2

Multiply $4$ by $- 1$ .

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 \int x^{- 4} d x + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 9

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

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 (- \frac{1}{3} x^{- 3} + C) + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 10

Simplify.

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Step 10.1

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

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 (- \frac{x^{- 3}}{3} + C) + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 10.2

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

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 (- \frac{1}{3 x^{3}} + C) + \int 6 x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 11

Since $6$ is constant with respect to $x$ , move $6$ out of the integral.

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 (- \frac{1}{3 x^{3}} + C) + 6 \int x d x + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 12

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

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 (- \frac{1}{3 x^{3}} + C) + 6 (\frac{1}{2} x^{2} + C) + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 13

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

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 (- \frac{1}{3 x^{3}} + C) + 6 (\frac{x^{2}}{2} + C) + \int 2 \cos (x) d x + \int \ln (5) d x$

Step 14

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

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 (- \frac{1}{3 x^{3}} + C) + 6 (\frac{x^{2}}{2} + C) + 2 \int \cos (x) d x + \int \ln (5) d x$

Step 15

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

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 (- \frac{1}{3 x^{3}} + C) + 6 (\frac{x^{2}}{2} + C) + 2 (\sin (x) + C) + \int \ln (5) d x$

Step 16

Apply the constant rule.

$2 (\frac{9 x^{\frac{11}{9}}}{11} + C) - 2 (- \frac{1}{3 x^{3}} + C) + 6 (\frac{x^{2}}{2} + C) + 2 (\sin (x) + C) + \ln (5) x + C$

Step 17

Simplify.

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Step 17.1

Simplify.

$\frac{18 x^{\frac{11}{9}}}{11} + \frac{2}{3 x^{3}} + 3 x^{2} + 2 \sin (x) + \ln (5) x + C$

Step 17.2

Reorder terms.

$\frac{18}{11} x^{\frac{11}{9}} + \frac{2}{3 x^{3}} + 3 x^{2} + 2 \sin (x) + \ln (5) x + C$

Step 18

The answer is the antiderivative of the function $f (x) = 2 x^{\frac{2}{9}} - \frac{2}{x^{4}} + 6 x + 2 \cos (x) + \ln (5)$ .

$F (x) =$ $\frac{18}{11} x^{\frac{11}{9}} + \frac{2}{3 x^{3}} + 3 x^{2} + 2 \sin (x) + \ln (5) x + C$