Calculus Examples

$\frac{2 x + 3}{x^{2}}$

Step 1

Write $\frac{2 x + 3}{x^{2}}$ as a function.

$f (x) = \frac{2 x + 3}{x^{2}}$

Step 2

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 3

Set up the integral to solve.

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

Step 4

Apply basic rules of exponents.

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Step 4.1

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

$\int (2 x + 3) {(x^{2})}^{- 1} d x$

Step 4.2

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

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Step 4.2.1

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

$\int (2 x + 3) x^{2 \cdot - 1} d x$

Step 4.2.2

Multiply $2$ by $- 1$ .

$\int (2 x + 3) x^{- 2} d x$

Step 5

Multiply $(2 x + 3) x^{- 2}$ .

$\int 2 x \cdot x^{- 2} + 3 x^{- 2} d x$

Step 6

Multiply $x$ by $x^{- 2}$ by adding the exponents.

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Step 6.1

Move $x^{- 2}$ .

$\int 2 (x^{- 2} x) + 3 x^{- 2} d x$

Step 6.2

Multiply $x^{- 2}$ by $x$ .

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Step 6.2.1

Raise $x$ to the power of $1$ .

$\int 2 (x^{- 2} x^{1}) + 3 x^{- 2} d x$

Step 6.2.2

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

$\int 2 x^{- 2 + 1} + 3 x^{- 2} d x$

Step 6.3

Add $- 2$ and $1$ .

$\int 2 x^{- 1} + 3 x^{- 2} d x$

Step 7

Split the single integral into multiple integrals.

$\int 2 x^{- 1} d x + \int 3 x^{- 2} d x$

Step 8

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

$2 \int x^{- 1} d x + \int 3 x^{- 2} d x$

Step 9

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

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify.

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Step 12.1

Simplify.

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

Step 12.2

Simplify.

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Step 12.2.1

Multiply $- 1$ by $3$ .

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

Step 12.2.2

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

$2 \ln (| x |) + \frac{- 3}{x} + C$

Step 12.2.3

Move the negative in front of the fraction.

$2 \ln (| x |) - \frac{3}{x} + C$

Step 13

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

$F (x) =$ $2 \ln (| x |) - \frac{3}{x} + C$