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Applications (Word Problem)

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Question

Solve : $\int \frac{x^{3} d x}{(x^{2} + a^{2}) (x^{2} + b^{2})}$

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Solution

$\int \frac{x^{3}}{(x^{2} + a^{2}) (x^{2} + b^{2})} d x$

Upon partial fraction decomposition,

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

Upon applying linearity

$= \frac{b^{2}}{b^{2} - a^{2}} \int \frac{x}{x^{2} + b^{2}} d x - \frac{a^{2}}{b^{2} - a^{2}} \int \frac{x}{x^{2} + a^{2}} d x$

$u = x^{2} + b^{2} \to d u = 2 x d x$

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

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

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\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 =

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

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Applications (Word Problem)

Standard VIII Mathematics

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