Mathematics

Evaluate the following definite integrals :
$$\displaystyle \int _{0}^{\pi} \dfrac {1}{1+\sin x}dx$$


SOLUTION
$$I=\displaystyle \int _{0}^{\pi} \dfrac {1}{1+\sin x}dx$$

$$=\displaystyle \int _{0}^{\pi}\dfrac {1-\sin x}{(1+\sin x)(1-\sin x)}dx$$

$$ =\displaystyle \int _{0}^{\pi} \dfrac {1-\sin x}{\cos^2 x}dx$$

$$\Rightarrow \ I=\displaystyle \int _{0}^{\pi} (\sec^2 x-\sec x\tan x)dx $$

$$= [\tan x-\sec x]_{0}^{\pi}$$

$$\Rightarrow \ I=(\tan \pi -\sec \pi )-(\tan 0-\sec 0)=0+1+1=2$$    
View Full Answer

Its FREE, you're just one step away


Subjective Medium Published on 17th 09, 2020
Next Question
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86
Enroll Now For FREE

Realted Questions

Q1 Subjective Medium
Solve:
$$\displaystyle \int_{\dfrac {-\pi}{2}}^{\dfrac{\pi}{2}} {\sin 5x.dx }$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Hard
Find : $$\displaystyle \int_{0}^{\tfrac {\pi}{2}} \dfrac {dx}{4 + 5\cos x} dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
$$\displaystyle\int_{0}^{\dfrac{\pi }{2}}\frac{\sqrt{\sin x}}{\sqrt{\sin x}+\sqrt{\cos x}}dx$$       

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Evaluate $$\int \dfrac{\sec^2 x}{3+\tan x}dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Hard
In calculating a number of integrals we had to use the method of integration by parts several times in succession.
The result could be obtained more rapidly and in a more concise form by using the so-called generalized formula for integration by parts
$$\displaystyle \int u\left ( x \right )v\left ( x \right )dx=u\left ( x \right )v_{1}-u'\left ( x \right )v_{2}\left ( x \right )+u''\left ( x \right )v_{3}\left ( x \right )+...+\left ( -1 \right )^{n-1}u^{n-1}\left ( x \right )V_{n}\left ( x \right ) \\ -\left ( -1 \right )^{n-1}\int u^{n}\left ( x \right )V_{n}\left ( x \right )dx$$ 
where  $$\displaystyle v_{1}\left ( x \right )=\int v\left ( x \right )dx,v_{2}\left ( x \right )=\int v_{1}\left ( x \right )dx ..., v_{n}\left ( x \right )= \int v_{n-1}\left ( x \right )dx$$
Of course, we assume that all derivatives and integrals appearing in this formula exist. The use of the generalized formula for integration by parts is especially useful when  calculating $$\displaystyle \int P_{n}\left ( x \right )Q\left ( x \right )dx,$$ where $$\displaystyle P_{n}\left ( x \right )$$ is polynomial of degree n and the factor Q(x) is such that it can be integrated successively n+1 times.

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer