What is the integral of 4/(sqrt(x+1)+\sqrt{x)} ?

The integral of 4/(sqrt(x+1)+\sqrt{x)} is 2(sqrt(x)+1/(3(sqrt(x)+\sqrt{x+1))^3}+sqrt(x+1))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of 4/(sqrt(x+1)+\sqrt{x)}

\int \frac{4}{x + 1 + x} d x

2 (x + \frac{1}{3 ( x + x + 1 ) ^{3}} + x + 1) + C

Solution steps

\int \frac{4}{x + 1 + x} d x

Take the constant out:

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

= 4 \cdot \int \frac{1}{x + 1 + x} d x

Apply u-substitution

: \int \frac{( u + 1 ) ( u - 1 ) ( u ^{2} + 1 )}{2 u ^{4}} d u

= 4 \cdot \int \frac{( u + 1 ) ( u - 1 ) ( u ^{2} + 1 )}{2 u ^{4}} d u

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \int \frac{( u + 1 ) ( u - 1 ) ( u ^{2} + 1 )}{u ^{4}} d u

Expand

\frac{( u + 1 ) ( u - 1 ) ( u ^{2} + 1 )}{u ^{4}} : 1 - \frac{1}{u ^{4}}

= 4 \cdot \frac{1}{2} \cdot \int 1 - \frac{1}{u ^{4}} d u

Apply the Sum Rule:

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

= 4 \cdot \frac{1}{2} (\int 1 d u - \int \frac{1}{u ^{4}} d u)

\int 1 d u = u

\int \frac{1}{u ^{4}} d u = - \frac{1}{3 u ^{3}}

= 4 \cdot \frac{1}{2} (u - (- \frac{1}{3 u ^{3}}))

Substitute back

u = x + 1 + x

= 4 \cdot \frac{1}{2} (x + 1 + x - (- \frac{1}{3 ( x + 1 + x ) ^{3}}))

Simplify

4 \cdot \frac{1}{2} (x + 1 + x - (- \frac{1}{3 ( x + 1 + x ) ^{3}})) : 2 (x + \frac{1}{3 ( x + x + 1 ) ^{3}} + x + 1)

= 2 (x + \frac{1}{3 ( x + x + 1 ) ^{3}} + x + 1)

Add a constant to the solution

= 2 (x + \frac{1}{3 ( x + x + 1 ) ^{3}} + x + 1) + C

\int \frac{x ^{3} + 4 x - 3}{x} d x

\int \frac{y ^{2}}{1} d y

\int 4 y e^{8 y^{2}} d y

\int \frac{( sin ( x ) )}{cos ^{3} ( x )} d x

\int 12 + \frac{500}{x + 1} d x

What is the integral of 4/(sqrt(x+1)+\sqrt{x)} ?
The integral of 4/(sqrt(x+1)+\sqrt{x)} is 2(sqrt(x)+1/(3(sqrt(x)+\sqrt{x+1))^3}+sqrt(x+1))+C