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# Integrate This

Ive been struggling with this for days. Guys, enlighten me!

Note by Danny Kills
3 years, 5 months ago

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Let

$$I(a)=\displaystyle\int_{0}^{\infty} \frac{\ln (1+a^{2}x^{2})}{1+b^{2}x^{2}}\text{ }\text{d}x$$

Differentiating w.r.t. $$a$$:

$$I'(a)=\displaystyle\int_{0}^{\infty} \frac{2ax^{2}}{(1+a^{2}x^{2})(1+b^{2}x^{2})}\text{ }\text{d}x$$

$$\therefore I'(a)=\displaystyle\frac{\pi}{b(b+a)}$$

$$\therefore I(a)=\dfrac{\pi}{b}\ln (a+b)+\mathcal{C}$$

Substituting $$a=0$$ we obtain the value of $$\mathcal{C}$$ as $$-\dfrac{\pi}{b}\ln b$$.

- 3 years, 5 months ago

A really elegant solution. Thanks!

- 3 years, 5 months ago