What is the integral of (sqrt(x+1))/(x^{5/2)} ?

The integral of (sqrt(x+1))/(x^{5/2)} is -(2(x+1)^{3/2})/(3x^{3/2)}+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (sqrt(x+1))/(x^{5/2)}

\int \frac{x + 1}{x ^{\frac{5}{2}}} d x

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

Solution steps

\int \frac{x + 1}{x ^{\frac{5}{2}}} d x

Apply u-substitution

= \int \frac{2 u ^{2}}{( u ^{2} - 1 ) ^{\frac{5}{2}}} d u

Take the constant out:

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

= 2 \cdot \int \frac{u ^{2}}{( u ^{2} - 1 ) ^{\frac{5}{2}}} d u

Apply Trigonometric Substitution

= 2 \cdot \int cot^{4} (v) sec^{3} (v) d v

Rewrite using trig identities

= 2 \cdot \int \frac{cos ( v )}{sin ^{4} ( v )} d v

Apply u-substitution

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

Apply the Power Rule

= 2 (- \frac{1}{3 w ^{3}})

Substitute back

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

Simplify

2 (- \frac{1}{3 sin ^{3} ( arcsec ( x + 1 ) )}) : - \frac{2 ( x + 1 ) ^{\frac{3}{2}}}{3 x ^{\frac{3}{2}}}

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

Add a constant to the solution

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

\int \frac{1}{4 + 4 sin ( x )} d x

\int \frac{ln ( x )}{x ^{13}} d x

\int 4 e^{- 2 x} + 5 e^{3 x} d x

\int (\frac{x}{x ^{2} - 9}) d x

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

What is the integral of (sqrt(x+1))/(x^{5/2)} ?
The integral of (sqrt(x+1))/(x^{5/2)} is -(2(x+1)^{3/2})/(3x^{3/2)}+C