Question:
$ \int\limits_{0} ^{\pi/2}\frac{\cos\,2x\,dx}{(\sin\,x+\cos\,x)^2}$ is equal to
$ \int\limits_{0} ^{\pi/2}\frac{\cos\,2x\,dx}{(\sin\,x+\cos\,x)^2}$ is equal to
Updated On: Jul 6, 2022
- 0
- $\frac{\pi}{4}$
- $\frac{\pi}{2}$
- none of these.
Hide Solution
Verified By Collegedunia
The Correct Option is A
Solution and Explanation
$\int\limits_{0}^{\pi /2} \frac{cos\,2x}{\left(sin\,x+cos\,x\right)^{2}} dx$
$=\int\limits_{0}^{\pi /2} \frac{cos^{2}\,x-sin^{2}\,x}{\left(sin\,x+cos\,x\right)^{2}}dx$
$=\int\limits_{0}^{\pi/ 2}\frac{cos\,x-sin\,x}{cos\,x+sin\,x}dx=\left|log \left(cos\,x+sin\,x\right)\right|_{0}^{\pi/ 2}$
$=log\,1-log\,1=0$
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 order and degree of the differential equation \( \sqrt{\frac{dy}{dx}} - 4 \frac{dy}{dx} - 7x = 0 \) are respectively
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.