Mathematics

# Evaluate by using properies of definite integrals $\int_{0}^{\pi }\frac{xdx}{a^{2}cos^{2}x+b^{2}sin^{2}x}$

##### SOLUTION
$I=\int^\pi_0\cfrac{x}{a^2\cos^2 x+b^2\sin^2 x}dx$
$\implies$ $I=\int^\pi_0\cfrac{\pi-x}{a^2\cos^2 x+b^2\sin^2 x}dx$
Adding above two equations $I+I=\int^\pi_0\cfrac{x}{a^2\cos^2 x+b^2\sin^2 x}dx$$+\int^\pi_0\cfrac{\pi-x}{a^2\cos^2 x+b^2\sin^2 x}dx$
$2I=\int^\pi_0\cfrac{\pi}{a^2\cos^2 x+b^2\sin^2 x}dx$
$2I=\int^\pi_0\cfrac{\pi}{a^2\cos^2 x+b^2\sin^2 x}dx$
$I=\cfrac{\pi}{2}\int^\pi_2\cfrac{sec^{2} x}{a^2+b^2\tan^2x}$
Now, let $tan$ $x$ $=$ $t$
Therefore,
$sec^{2} x dx = dt$
$I=\dfrac{2\pi}{2}\int^\infty_{0}\dfrac{dt}{a^{2} + b^{2}t^{2}}$
$I=\dfrac{\pi}{b^{2}}\int^\infty_{0}\dfrac{dt}{(\dfrac{a}{b})^{2} + t^{2}}$
$I=\dfrac{\pi}{b^{2}}\times \dfrac{b}{a} [tan^{-1}\dfrac{at}{b}]^\infty_{0}$
$I=\dfrac{\pi}{ab}\times \dfrac{\pi}{2}$
$I=\dfrac{\pi^{2}}{2ab}$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Subjective Hard
Express $\int_{0}^{2}e^{x} dx$ as the limit of a sum.

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Hard
If $\displaystyle \int_{0}^{x}[x]dx=\displaystyle \int_{0}^{[x]}xdx,x \notin$ integer (where,[.] and {.} denotes the greatest integer and fractional parts respectively.then the value of 4{x} is equal to ...

1 Verified Answer | Published on 17th 09, 2020

Q3 One Word Medium
Evaluate:$\displaystyle\int \frac{dx}{\sqrt{1+4x^{2}}}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle\int \frac{3x+4}{x^{2}+4x+2}dx.$
• A. $\displaystyle \frac{3}{2}\log \left ( x^{2}+4x+2 \right )+\frac{1}{\sqrt{\left ( 2 \right )}}\log \frac{x+2-\sqrt{2}}{x+2+\sqrt{\left ( 2 \right )}}.$
• B. $\displaystyle \frac{3}{2}\log \left ( x^{2}+4x+2 \right )-\frac{1}{2\sqrt{\left ( 2 \right )}}\log \frac{x+2-\sqrt{2}}{x+2+\sqrt{\left ( 2 \right )}}.$
• C. $\displaystyle \frac{3}{2}\log \left ( x^{2}+4x+2 \right )-\frac{1}{\sqrt{2\left ( 2 \right )}}\log \frac{x+2-2\sqrt{2}}{x+2+\sqrt{\left ( 2 \right )}}.$
• D. $\displaystyle \frac{3}{2}\log \left ( x^{2}+4x+2 \right )-\frac{1}{\sqrt{\left ( 2 \right )}}\log \frac{x+2-\sqrt{2}}{x+2+\sqrt{\left ( 2 \right )}}.$

1 Verified Answer | Published on 17th 09, 2020

Q5 Subjective Medium
If $y=2^23^{2x}5^{-5}7^{-5}$ then $\dfrac{dy}{dx}=$