Question:
$\int\frac{dx}{1-cos x -sin x} $ is equal to
$\int\frac{dx}{1-cos x -sin x} $ is equal to
Updated On: Jul 6, 2022
- $log \left|1+cot \frac{x}{2}\right| +C$
- $log \left|1-tan \frac{x}{2}\right| +C$
- $log \left|1-cot \frac{x}{2}\right| +C$
- $log \left|1+tan \frac{x}{2}\right| +C$
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The Correct Option is C
Solution and Explanation
Let$ I = \int \frac{dx}{1-cos x -sin x}$
put $cos x = \frac{1-tan^{2} \frac{x}{2}}{1+tan^{2} \frac{x}{2}} and sin x= \frac{2tan \frac{x}{2}}{1+tan ^{2 } \frac{x}{2}}$
$ \therefore I= \int\frac{dx}{1-\left(\frac{1-tan^{2} \frac{x}{2}}{1+tan^{2} \frac{x}{2}}\right)- \frac{2tan \frac{x}{2}}{1+tan ^{ 2} \frac{x}{2}}} $
$= \int \frac{sec ^{2} \frac{x}{2} dx }{2tan ^{2} \frac{x}{2}-2tan \frac{x}{2}} = \int\frac{\frac{1}{2}sec^{2} \frac{x}{2} dx }{tan^{2} \frac{x}{2} -tan \frac{x}{2}} $
Put $tan \frac{x}{2}=t \Rightarrow \frac{1}{2} sec^{2} \frac{x}{2 }dx = dt $
$\therefore I = \int \frac{dt}{t^{2} -t} = \left[\frac{1}{t-1}-\frac{1}{t}\right]\int+C $
$=log\left(t-1\right) -logt + C = log\left|\frac{t-1}{t} +C\right| $
$=log \left|1-cot \frac{x}{2}\right| +C$
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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.