What is the integral of (3y)/(sqrt(2y^2+5)) ?

The integral of (3y)/(sqrt(2y^2+5)) is 3/2 sqrt(2y^2+5)+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral of (3y)/(sqrt(2y^2+5))

\int \frac{3 y}{2 y ^{2} + 5} d y

\frac{3}{2} 2 y^{2} + 5 + C

Solution steps

\int \frac{3 y}{2 y ^{2} + 5} d y

Take the constant out:

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

= 3 \cdot \int \frac{y}{2 y ^{2} + 5} d y

Apply u-substitution

= 3 \cdot \int \frac{1}{4 u} d u

Take the constant out:

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

= 3 \cdot \frac{1}{4} \cdot \int \frac{1}{u} d u

Apply the Power Rule

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

Substitute back

u = 2 y^{2} + 5

= 3 \cdot \frac{1}{4} \cdot 2 (2 y^{2} + 5)^{\frac{1}{2}}

Simplify

3 \cdot \frac{1}{4} \cdot 2 (2 y^{2} + 5)^{\frac{1}{2}} : \frac{3}{2} 2 y^{2} + 5

= \frac{3}{2} 2 y^{2} + 5

Add a constant to the solution

= \frac{3}{2} 2 y^{2} + 5 + C

What is the integral of (3y)/(sqrt(2y^2+5)) ?
The integral of (3y)/(sqrt(2y^2+5)) is 3/2 sqrt(2y^2+5)+C