Calculus Examples

$\frac{x + 1}{x^{2}}$

Step 1

Write $\frac{x + 1}{x^{2}}$ as a function.

$f (x) = \frac{x + 1}{x^{2}}$

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.

$F (x) = \int \frac{x + 1}{x^{2}} d x$

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.

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

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

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

Step 4.2.2

Multiply $2$ by $- 1$ .

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

Step 5

Multiply $(x + 1) x^{- 2}$ .

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

Step 6

Simplify.

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Step 6.1

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

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Step 6.1.1

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

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Step 6.1.1.1

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

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

Step 6.1.1.2

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

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

Step 6.1.2

Subtract $2$ from $1$ .

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

Step 6.2

Multiply $x^{- 2}$ by $1$ .

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

Step 7

Split the single integral into multiple integrals.

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

Step 8

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

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

Step 9

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

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

Step 10

Simplify.

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

Step 11

The answer is the antiderivative of the function $f (x) = \frac{x + 1}{x^{2}}$ .

$F (x) =$ $\ln (| x |) - \frac{1}{x} + C$