MyQuestionIcon

1

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Particular Solution of a Differential Equation

Solve : ∫x ...

Question

Solve : $\int \frac{x d x}{(x^{2} + a^{2}) (x^{2} + b^{2})}$

Open in App

Solution

$\int \frac{x}{(x^{2} + a^{2}) (x^{2} + b^{2})} d x$

substitute $u = \frac{x^{2}}{2} \to d x = \frac{1}{x} d u$

$= \int \frac{1}{(2 u + a^{2}) (2 u + b^{2})} d u$

Upon partial fraction decomposition

$= \int (\frac{1}{(b^{2} - a^{2}) (2 u + a^{2})} - \frac{1}{(b^{2} - a^{2}) (2 u + b^{2})}) d u$

upon linearity

$= \frac{1}{(b^{2} - a^{2})} \int \frac{1}{2 u + a^{2}} d u - \frac{1}{(b^{2} - a^{2})} \int \frac{1}{2 u + b^{2}} d u$

$= \frac{1}{(b^{2} - a^{2})} \frac{ln (2 u + a^{2})}{2} - \frac{1}{(b^{2} - a^{2})} \frac{ln (2 u + b^{2})}{2}$

$= - \frac{ln (x^{2} + b^{2}) - ln (x^{2} + a^{2})}{2 (b^{2} - a^{2})} + C$

Suggest Corrections

thumbs-up

0

Similar questions

\int \frac{x d x}{(x^{2} + a^{2}) (x^{2} + b^{2})}

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

\int_{0}^{\infty} \frac{1}{(x^{2} + a^{2}) (x^{2} + b^{2})} d x

Q. Solve the following quadratic equation using quadratic formula:

a^{2} x^{2} + (a^{2} - b^{2}) x - b^{2} = 0

\frac{x^{2}}{(x^{2} + a^{2}) (x^{2} + b^{2})} = k [\frac{a^{2}}{x^{2} + a^{2}} - \frac{b^{2}}{x^{2} + b^{2}}] \Rightarrow k =

View More

Join BYJU'S Learning Program

similar_icon

Related Videos

lock

General and Particular Solutions of a DE

MATHEMATICS

Watch in App

explore_more_icon

Explore more

Particular Solution of a Differential Equation

Standard XII Mathematics

Join BYJU'S Learning Program

CrossIcon