What is the integral of (3ax)/(b^2+c^2x^2) ?

The integral of (3ax)/(b^2+c^2x^2) is a 3/(2c^2)ln|b^2+c^2x^2|+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of (3ax)/(b^2+c^2x^2)

\int \frac{3 a x}{b ^{2} + c ^{2} x ^{2}} d x

a \frac{3}{2 c ^{2}} ln b^{2} + c^{2} x^{2} + C

Solution steps

\int \frac{3 a x}{b ^{2} + c ^{2} x ^{2}} d x

Take the constant out:

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

= 3 a \cdot \int \frac{x}{b ^{2} + c ^{2} x ^{2}} d x

Apply u-substitution

: \int \frac{1}{2 c ^{2} u} d u

= 3 a \cdot \int \frac{1}{2 c ^{2} u} d u

Take the constant out:

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

= 3 a \frac{1}{2 c ^{2}} \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 3 a \frac{1}{2 c ^{2}} ln ∣ u ∣

Substitute back

u = b^{2} + c^{2} x^{2}

= 3 a \frac{1}{2 c ^{2}} ln b^{2} + c^{2} x^{2}

Simplify

3 a \frac{1}{2 c ^{2}} ln b^{2} + c^{2} x^{2} : a \frac{3}{2 c ^{2}} ln b^{2} + c^{2} x^{2}

= a \frac{3}{2 c ^{2}} ln b^{2} + c^{2} x^{2}

Add a constant to the solution

= a \frac{3}{2 c ^{2}} ln b^{2} + c^{2} x^{2} + C

y^{^{'}} = 4 x - y

\int s 3^{s} d s

\int e^{- θ} cos (4 θ) d θ

\frac{\partial}{\partial x} (2 x + 3)

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

What is the integral of (3ax)/(b^2+c^2x^2) ?
The integral of (3ax)/(b^2+c^2x^2) is a 3/(2c^2)ln|b^2+c^2x^2|+C