Calculus Examples

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

Step 1

Let $u_{2} = e^{x} + e^{- x}$ . Then $d u_{2} = (e^{x} - e^{- x}) d x$ , so $\frac{1}{e^{x} - e^{- x}} d u_{2} = d x$ . Rewrite using $u_{2}$ and $d$ $u_{2}$ .

Tap for more steps...

Step 1.1

Let $u_{2} = e^{x} + e^{- x}$ . Find $\frac{d u_{2}}{d x}$ .

Tap for more steps...

Step 1.1.1

Differentiate $e^{x} + e^{- x}$ .

$\frac{d}{d x} [e^{x} + e^{- x}]$

Step 1.1.2

By the Sum Rule, the derivative of $e^{x} + e^{- x}$ with respect to $x$ is $\frac{d}{d x} [e^{x}] + \frac{d}{d x} [e^{- x}]$ .

$\frac{d}{d x} [e^{x}] + \frac{d}{d x} [e^{- x}]$

Step 1.1.3

Differentiate using the Exponential Rule which states that $\frac{d}{d x} [a^{x}]$ is $a^{x} \ln (a)$ where $a$ = $e$ .

$e^{x} + \frac{d}{d x} [e^{- x}]$

Step 1.1.4

Evaluate $\frac{d}{d x} [e^{- x}]$ .

Tap for more steps...

Step 1.1.4.1

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = e^{x}$ and $g (x) = - x$ .

Tap for more steps...

Step 1.1.4.1.1

To apply the Chain Rule, set $u_{1}$ as $- x$ .

$e^{x} + \frac{d}{d u_{1}} [e^{u_{1}}] \frac{d}{d x} [- x]$

Step 1.1.4.1.2

Differentiate using the Exponential Rule which states that $\frac{d}{d u_{1}} [a^{u_{1}}]$ is $a^{u_{1}} \ln (a)$ where $a$ = $e$ .

$e^{x} + e^{u_{1}} \frac{d}{d x} [- x]$

Step 1.1.4.1.3

Replace all occurrences of $u_{1}$ with $- x$ .

$e^{x} + e^{- x} \frac{d}{d x} [- x]$

Step 1.1.4.2

Since $- 1$ is constant with respect to $x$ , the derivative of $- x$ with respect to $x$ is $- \frac{d}{d x} [x]$ .

$e^{x} + e^{- x} (- \frac{d}{d x} [x])$

Step 1.1.4.3

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

$e^{x} + e^{- x} (- 1 \cdot 1)$

Step 1.1.4.4

Multiply $- 1$ by $1$ .

$e^{x} + e^{- x} \cdot - 1$

Step 1.1.4.5

Move $- 1$ to the left of $e^{- x}$ .

$e^{x} - 1 \cdot e^{- x}$

Step 1.1.4.6

Rewrite $- 1 e^{- x}$ as $- e^{- x}$ .

$e^{x} - e^{- x}$

Step 1.2

Rewrite the problem using $u_{2}$ and $d u_{2}$ .

$\int \frac{1}{{u_{2}}^{\frac{7}{2}}} d u_{2}$

Step 2

Apply basic rules of exponents.

Tap for more steps...

Step 2.1

Move ${u_{2}}^{\frac{7}{2}}$ out of the denominator by raising it to the $- 1$ power.

$\int {({u_{2}}^{\frac{7}{2}})}^{- 1} d u_{2}$

Step 2.2

Multiply the exponents in ${({u_{2}}^{\frac{7}{2}})}^{- 1}$ .

Tap for more steps...

Step 2.2.1

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

$\int {u_{2}}^{\frac{7}{2} \cdot - 1} d u_{2}$

Step 2.2.2

Multiply $\frac{7}{2} \cdot - 1$ .

Tap for more steps...

Step 2.2.2.1

Combine $\frac{7}{2}$ and $- 1$ .

$\int {u_{2}}^{\frac{7 \cdot - 1}{2}} d u_{2}$

Step 2.2.2.2

Multiply $7$ by $- 1$ .

$\int {u_{2}}^{\frac{- 7}{2}} d u_{2}$

Step 2.2.3

Move the negative in front of the fraction.

$\int {u_{2}}^{- \frac{7}{2}} d u_{2}$

Step 3

By the Power Rule, the integral of ${u_{2}}^{- \frac{7}{2}}$ with respect to $u_{2}$ is $- \frac{2}{5} {u_{2}}^{- \frac{5}{2}}$ .

$- \frac{2}{5} {u_{2}}^{- \frac{5}{2}} + C$

Step 4

Simplify.

Tap for more steps...

Step 4.1

Rewrite $- \frac{2}{5} {u_{2}}^{- \frac{5}{2}} + C$ as $- \frac{2}{5} \cdot \frac{1}{{u_{2}}^{\frac{5}{2}}} + C$ .

$- \frac{2}{5} \cdot \frac{1}{{u_{2}}^{\frac{5}{2}}} + C$

Step 4.2

Simplify.

Tap for more steps...

Step 4.2.1

Multiply $\frac{1}{{u_{2}}^{\frac{5}{2}}}$ by $\frac{2}{5}$ .

$- \frac{2}{{u_{2}}^{\frac{5}{2}} \cdot 5} + C$

Step 4.2.2

Move $5$ to the left of ${u_{2}}^{\frac{5}{2}}$ .

$- \frac{2}{5 {u_{2}}^{\frac{5}{2}}} + C$

Step 5

Replace all occurrences of $u_{2}$ with $e^{x} + e^{- x}$ .

$- \frac{2}{5 {(e^{x} + e^{- x})}^{\frac{5}{2}}} + C$