Question:
$\int\limits _{0}^{1} \frac {8 \log (1+x)}{1+x^2}dx$ is equal to
$\int\limits _{0}^{1} \frac {8 \log (1+x)}{1+x^2}dx$ is equal to
Updated On: Jul 6, 2022
- $\pi log 2$
- $\frac {\pi} {8} log 2$
- $\frac {\pi} {2} log 2$
- log 2.
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
Let $ I=\int\limits_{0}^{1} \frac{8}{1+x^{2}} log \left(1 + x \right)dx$. Put $x = tan \, \theta$
$\therefore I=8 \int\limits_{0}^{\pi /4} \frac{log\left(1+tan\,\theta\right)sec^{2}\,\theta}{1+tan^{2}\,\theta}$
$=8 \int\limits_{0}^{\pi /4}log\left(1+tan\,\theta\right)d\theta$
Also , $I=8 \int\limits_{\pi /4}^{\pi /4} log \left[1+tan\left(\frac{\pi}{4}-\theta\right)\right]d \theta$
$=8 \int\limits_{0}^{\pi /4} log\left(1+\frac{1-tan\,\theta}{1+tan\,\theta}\right)d \theta$
$=8 \int\limits_{0}^{\pi/ 4} log\left(\frac{2}{1+tan\,\theta}\right)dx$
$=8\left[log\,2 \int\limits_{0}^{\pi /4} d\theta\right] \Rightarrow2I $
$=log\,2\cdot\left(\frac{\pi}{4}\right)8$
$\Rightarrow I=\pi \, log\,2=2\pi\, log\,2$
Was this answer helpful?
0
0
Top Questions on integral
- The value of the integral \( \int_0^1 x^2 \, dx \) is:
- The value of the definite integral \( \int_0^{\pi} \sin^2 x \, dx \) is:
- Let $ f(x) $ be a positive function and $I_1 = \int_{-\frac{1}{2}}^1 2x \, f\left(2x(1-2x)\right) dx$ and $I_2 = \int_{-1}^2 f\left(x(1-x)\right) dx.$ Then the value of $\frac{I_2}{I_1}$ is equal to ____
The integral \(\int e^x \sqrt{e^x} \, dx\) equals:
The value of : \( \int \frac{x + 1}{x(1 + xe^x)} dx \).
View More Questions
Concepts Used:
Integral
The representation of the area of a region under a curve is called to be as integral. The actual value of an integral can be acquired (approximately) by drawing rectangles.
- The definite integral of a function can be shown as the area of the region bounded by its graph of the given function between two points in the line.
- The area of a region is found by splitting it into thin vertical rectangles and applying the lower and the upper limits, the area of the region is summarized.
- An integral of a function over an interval on which the integral is described.
Also, F(x) is known to be a Newton-Leibnitz integral or antiderivative or primitive of a function f(x) on an interval I.
F'(x) = f(x)
For every value of x = I.
Types of Integrals:
Integral calculus helps to resolve two major types of problems:
- The problem of getting a function if its derivative is given.
- The problem of getting the area bounded by the graph of a function under given situations.