Calculus Examples

$\frac{6}{x^{3}} - 4 e^{2 x} + 7$

Step 1

Write $\frac{6}{x^{3}} - 4 e^{2 x} + 7$ as a function.

$f (x) = \frac{6}{x^{3}} - 4 e^{2 x} + 7$

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.

$F (x) = \int \frac{6}{x^{3}} - 4 e^{2 x} + 7 d x$

Step 4

Split the single integral into multiple integrals.

$\int \frac{6}{x^{3}} d x + \int - 4 e^{2 x} d x + \int 7 d x$

Step 5

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

$6 \int \frac{1}{x^{3}} d x + \int - 4 e^{2 x} d x + \int 7 d x$

Step 6

Apply basic rules of exponents.

Tap for more steps...

Step 6.1

Move $x^{3}$ out of the denominator by raising it to the $- 1$ power.

$6 \int {(x^{3})}^{- 1} d x + \int - 4 e^{2 x} d x + \int 7 d x$

Step 6.2

Multiply the exponents in ${(x^{3})}^{- 1}$ .

Tap for more steps...

Step 6.2.1

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

$6 \int x^{3 \cdot - 1} d x + \int - 4 e^{2 x} d x + \int 7 d x$

Step 6.2.2

Multiply $3$ by $- 1$ .

$6 \int x^{- 3} d x + \int - 4 e^{2 x} d x + \int 7 d x$

Step 7

By the Power Rule, the integral of $x^{- 3}$ with respect to $x$ is $- \frac{1}{2} x^{- 2}$ .

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

Step 8

Simplify.

Tap for more steps...

Step 8.1

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

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

Step 8.2

Move $x^{- 2}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

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

Step 9

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

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

Step 10

Let $u = 2 x$ . Then $d u = 2 d x$ , so $\frac{1}{2} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 10.1

Let $u = 2 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 10.1.1

Differentiate $2 x$ .

$\frac{d}{d x} [2 x]$

Step 10.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

$2 \frac{d}{d x} [x]$

Step 10.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 1$ .

$2 \cdot 1$

Step 10.1.4

Multiply $2$ by $1$ .

$2$

Step 10.2

Rewrite the problem using $u$ and $d u$ .

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

Step 11

Combine $e^{u}$ and $\frac{1}{2}$ .

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

Step 12

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 13

Simplify.

Tap for more steps...

Step 13.1

Combine $\frac{1}{2}$ and $- 4$ .

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

Step 13.2

Cancel the common factor of $- 4$ and $2$ .

Tap for more steps...

Step 13.2.1

Factor $2$ out of $- 4$ .

$6 (- \frac{1}{2 x^{2}} + C) + \frac{2 \cdot - 2}{2} \int e^{u} d u + \int 7 d x$

Step 13.2.2

Cancel the common factors.

Tap for more steps...

Step 13.2.2.1

Factor $2$ out of $2$ .

$6 (- \frac{1}{2 x^{2}} + C) + \frac{2 \cdot - 2}{2 (1)} \int e^{u} d u + \int 7 d x$

Step 13.2.2.2

Cancel the common factor.

$6 (- \frac{1}{2 x^{2}} + C) + \frac{2 \cdot - 2}{2 \cdot 1} \int e^{u} d u + \int 7 d x$

Step 13.2.2.3

Rewrite the expression.

$6 (- \frac{1}{2 x^{2}} + C) + \frac{- 2}{1} \int e^{u} d u + \int 7 d x$

Step 13.2.2.4

Divide $- 2$ by $1$ .

$6 (- \frac{1}{2 x^{2}} + C) - 2 \int e^{u} d u + \int 7 d x$

Step 14

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$6 (- \frac{1}{2 x^{2}} + C) - 2 (e^{u} + C) + \int 7 d x$

Step 15

Apply the constant rule.

$6 (- \frac{1}{2 x^{2}} + C) - 2 (e^{u} + C) + 7 x + C$

Step 16

Simplify.

$- \frac{3}{x^{2}} - 2 e^{u} + 7 x + C$

Step 17

Replace all occurrences of $u$ with $2 x$ .

$- \frac{3}{x^{2}} - 2 e^{2 x} + 7 x + C$

Step 18

The answer is the antiderivative of the function $f (x) = \frac{6}{x^{3}} - 4 e^{2 x} + 7$ .

$F (x) =$ $- \frac{3}{x^{2}} - 2 e^{2 x} + 7 x + C$