Calculus Examples

$\int \frac{4}{x} - 2 e^{- 0.7 x} d x$

Step 1

Split the single integral into multiple integrals.

$\int \frac{4}{x} d x + \int - 2 e^{- 0.7 x} d x$

Step 2

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

$4 \int \frac{1}{x} d x + \int - 2 e^{- 0.7 x} d x$

Step 3

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

$4 (\ln (| x |) + C) + \int - 2 e^{- 0.7 x} d x$

Step 4

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

$4 (\ln (| x |) + C) - 2 \int e^{- 0.7 x} d x$

Step 5

Let $u = - 0.7 x$ . Then $d u = - 0.7 d x$ , so $\frac{1}{- 0.7} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

Tap for more steps...

Step 5.1

Let $u = - 0.7 x$ . Find $\frac{d u}{d x}$ .

Tap for more steps...

Step 5.1.1

Differentiate $- 0.7 x$ .

$\frac{d}{d x} [- 0.7 x]$

Step 5.1.2

Since $- 0.7$ is constant with respect to $x$ , the derivative of $- 0.7 x$ with respect to $x$ is $- 0.7 \frac{d}{d x} [x]$ .

$- 0.7 \frac{d}{d x} [x]$

Step 5.1.3

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

$- 0.7 \cdot 1$

Step 5.1.4

Multiply $- 0.7$ by $1$ .

$- 0.7$

Step 5.2

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

$4 (\ln (| x |) + C) - 2 \int e^{u} \frac{1}{- 0.7} d u$

Step 6

Simplify.

Tap for more steps...

Step 6.1

Move the negative in front of the fraction.

$4 (\ln (| x |) + C) - 2 \int e^{u} (- \frac{1}{0.7}) d u$

Step 6.2

Combine $e^{u}$ and $\frac{1}{0.7}$ .

$4 (\ln (| x |) + C) - 2 \int - \frac{e^{u}}{0.7} d u$

Step 7

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

$4 (\ln (| x |) + C) - 2 (- \int \frac{e^{u}}{0.7} d u)$

Step 8

Multiply $- 1$ by $- 2$ .

$4 (\ln (| x |) + C) + 2 \int \frac{e^{u}}{0.7} d u$

Step 9

Since $\frac{1}{0.7}$ is constant with respect to $u$ , move $\frac{1}{0.7}$ out of the integral.

$4 (\ln (| x |) + C) + 2 (\frac{1}{0.7} \int e^{u} d u)$

Step 10

Combine $\frac{1}{0.7}$ and $2$ .

$4 (\ln (| x |) + C) + \frac{2}{0.7} \int e^{u} d u$

Step 11

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

$4 (\ln (| x |) + C) + \frac{2}{0.7} (e^{u} + C)$

Step 12

Simplify.

$4 \ln (| x |) + 2.\overline{857142} e^{u} + C$

Step 13

Replace all occurrences of $u$ with $- 0.7 x$ .

$4 \ln (| x |) + 2.\overline{857142} e^{- 0.7 x} + C$