Calculus Examples

$f (x) = \frac{3}{x} + \frac{4}{x^{2}} - \frac{5}{\sin^{2} (x)}$

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.

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

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.

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

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, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 7.2.2

Multiply $2$ by $- 1$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Simplify.

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Step 11.1

Convert from $\frac{1}{\sin^{2} (x)}$ to $\csc^{2} (x)$ .

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

Step 11.2

Multiply $5$ by $- 1$ .

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

Step 12

Since the derivative of $- \cot (x)$ is $\csc^{2} (x)$ , the integral of $\csc^{2} (x)$ is $- \cot (x)$ .

$3 (\ln (| x |) + C) + 4 (- x^{- 1} + C) - 5 (- \cot (x) + C)$

Step 13

Simplify.

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Step 13.1

Simplify.

$3 \ln (| x |) - \frac{4}{x} - 5 (- \cot (x)) + C$

Step 13.2

Multiply $- 1$ by $- 5$ .

$3 \ln (| x |) - \frac{4}{x} + 5 \cot (x) + C$

Step 14

The answer is the antiderivative of the function $f (x) = \frac{3}{x} + \frac{4}{x^{2}} - \frac{5}{\sin^{2} (x)}$ .

$F (x) =$ $3 \ln (| x |) - \frac{4}{x} + 5 \cot (x) + C$