Calculus Examples

$f (x) = \frac{4}{x} + \frac{6}{\sqrt{x}} + \frac{7}{x^{2}}$

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.

$F (x) = \int \frac{4}{x} + \frac{6}{\sqrt{x}} + \frac{7}{x^{2}} d x$

Step 3

Split the single integral into multiple integrals.

$\int \frac{4}{x} d x + \int \frac{6}{\sqrt{x}} d x + \int \frac{7}{x^{2}} d x$

Step 4

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

$4 \int \frac{1}{x} d x + \int \frac{6}{\sqrt{x}} d x + \int \frac{7}{x^{2}} d x$

Step 5

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

$4 (\ln (| x |) + C) + \int \frac{6}{\sqrt{x}} d x + \int \frac{7}{x^{2}} d x$

Step 6

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

$4 (\ln (| x |) + C) + 6 \int \frac{1}{\sqrt{x}} d x + \int \frac{7}{x^{2}} d x$

Step 7

Apply basic rules of exponents.

Tap for more steps...

Step 7.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x}$ as $x^{\frac{1}{2}}$ .

$4 (\ln (| x |) + C) + 6 \int \frac{1}{x^{\frac{1}{2}}} d x + \int \frac{7}{x^{2}} d x$

Step 7.2

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

$4 (\ln (| x |) + C) + 6 \int {(x^{\frac{1}{2}})}^{- 1} d x + \int \frac{7}{x^{2}} d x$

Step 7.3

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

Tap for more steps...

Step 7.3.1

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

$4 (\ln (| x |) + C) + 6 \int x^{\frac{1}{2} \cdot - 1} d x + \int \frac{7}{x^{2}} d x$

Step 7.3.2

Combine $\frac{1}{2}$ and $- 1$ .

$4 (\ln (| x |) + C) + 6 \int x^{\frac{- 1}{2}} d x + \int \frac{7}{x^{2}} d x$

Step 7.3.3

Move the negative in front of the fraction.

$4 (\ln (| x |) + C) + 6 \int x^{- \frac{1}{2}} d x + \int \frac{7}{x^{2}} d x$

Step 8

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

$4 (\ln (| x |) + C) + 6 (2 x^{\frac{1}{2}} + C) + \int \frac{7}{x^{2}} d x$

Step 9

Since $7$ is constant with respect to $x$ , move $7$ out of the integral.

$4 (\ln (| x |) + C) + 6 (2 x^{\frac{1}{2}} + C) + 7 \int \frac{1}{x^{2}} d x$

Step 10

Apply basic rules of exponents.

Tap for more steps...

Step 10.1

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

$4 (\ln (| x |) + C) + 6 (2 x^{\frac{1}{2}} + C) + 7 \int {(x^{2})}^{- 1} d x$

Step 10.2

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

Tap for more steps...

Step 10.2.1

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

$4 (\ln (| x |) + C) + 6 (2 x^{\frac{1}{2}} + C) + 7 \int x^{2 \cdot - 1} d x$

Step 10.2.2

Multiply $2$ by $- 1$ .

$4 (\ln (| x |) + C) + 6 (2 x^{\frac{1}{2}} + C) + 7 \int x^{- 2} d x$

Step 11

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

$4 (\ln (| x |) + C) + 6 (2 x^{\frac{1}{2}} + C) + 7 (- x^{- 1} + C)$

Step 12

Simplify.

Tap for more steps...

Step 12.1

Simplify.

$4 \ln (| x |) + 12 x^{\frac{1}{2}} + 7 (- x^{- 1}) + C$

Step 12.2

Multiply $- 1$ by $7$ .

$4 \ln (| x |) + 12 x^{\frac{1}{2}} - 7 x^{- 1} + C$

Step 13

The answer is the antiderivative of the function $f (x) = \frac{4}{x} + \frac{6}{\sqrt{x}} + \frac{7}{x^{2}}$ .

$F (x) =$ $4 \ln (| x |) + 12 x^{\frac{1}{2}} - 7 x^{- 1} + C$