Waste less time on Facebook — follow Brilliant.
×

Integrate This

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

Note by Danny Kills
2 years, 10 months ago

No vote yet
1 vote

Comments

Sort by:

Top Newest

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 \(-\dfrac{\pi}{b}\ln b\). Karthik Kannan · 2 years, 9 months ago

Log in to reply

@Karthik Kannan A really elegant solution. Thanks! Danny Kills · 2 years, 9 months ago

Log in to reply

×

Problem Loading...

Note Loading...

Set Loading...