en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sqrt(2+e^x)

Solution

\int 2 + e^{x} d x

Solution

- 2 ln \frac{1}{2} (2 + e^{x}) + 1 + 2 ln \frac{1}{2} (2 + e^{x}) - 1 + 22 \frac{1}{2} (2 + e^{x}) + C

Solution steps

\int 2 + e^{x} d x

Apply u-substitution

: \int \frac{2 + u}{u} d u

= \int \frac{2 + u}{u} d u

Apply u-substitution

: \int \frac{2 v ^{2}}{v ^{2} - 2} d v

= \int \frac{2 v ^{2}}{v ^{2} - 2} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{v ^{2}}{v ^{2} - 2} d v

Apply Integral Substitution

= 2 \cdot \int \frac{2 w ^{2}}{w ^{2} - 1} d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 22 \cdot \int \frac{w ^{2}}{w ^{2} - 1} d w

Factor

w^{2} - 1 : - (- w^{2} + 1)

= 22 \cdot \int \frac{w ^{2}}{- ( - w ^{2} + 1 )} d w

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 22 (- \int \frac{w ^{2}}{- w ^{2} + 1} d w)

\frac{w ^{2}}{- w ^{2} + 1} = \frac{1}{- w ^{2} + 1} - 1

= 22 (- \int \frac{1}{- w ^{2} + 1} - 1 d w)

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 22 (- (\int \frac{1}{- w ^{2} + 1} d w - \int 1 d w))

\int \frac{1}{- w ^{2} + 1} d w = \frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2}

\int 1 d w = w

= 22 (- (\frac{ln ∣ w + 1 ∣}{2} - \frac{ln ∣ w - 1 ∣}{2} - w))

Substitute back

= 22 - \frac{ln \frac{2 + e ^{x}}{2} + 1}{2} - \frac{ln \frac{2 + e ^{x}}{2} - 1}{2} - \frac{2 + e ^{x}}{2}

Simplify

22 - \frac{ln \frac{2 + e ^{x}}{2} + 1}{2} - \frac{ln \frac{2 + e ^{x}}{2} - 1}{2} - \frac{2 + e ^{x}}{2} : - 2 ln \frac{1}{2} (2 + e^{x}) + 1 + 2 ln \frac{1}{2} (2 + e^{x}) - 1 + 22 \frac{1}{2} (2 + e^{x})

= - 2 ln \frac{1}{2} (2 + e^{x}) + 1 + 2 ln \frac{1}{2} (2 + e^{x}) - 1 + 22 \frac{1}{2} (2 + e^{x})

Add a constant to the solution

= - 2 ln \frac{1}{2} (2 + e^{x}) + 1 + 2 ln \frac{1}{2} (2 + e^{x}) - 1 + 22 \frac{1}{2} (2 + e^{x}) + C

Popular Examples

integral of (x^2+2x+2)/(x^3-1)

\int \frac{x ^{2} + 2 x + 2}{x ^{3} - 1} d x

integral of 1/(u^7)

\int \frac{1}{u ^{7}} d u

integral of 3x^4+x^2+2

\int 3 x^{4} + x^{2} + 2 d x

integral of 5x^2+x

\int 5 x^{2} + x d x

integral of (2x+3)/(x^2+2x+3)

\int \frac{2 x + 3}{x ^{2} + 2 x + 3} d x