Mathematics

$$\displaystyle \underset{0}{\overset{\pi}{\int}} \dfrac{x \, dx}{ 1 + \sin \, x}$$ =


ANSWER

$$\pi$$


SOLUTION
$$\int \dfrac{x}{1+\sin x}dx$$

$$=\int \dfrac{x}{\frac{\cos ^2x}{1-\sin x}}dx$$

$$=\int \left ( \dfrac{x}{\cos ^2x} -\dfrac{x \sin x}{\cos ^2x}\right )dx$$

$$=x \tan x +\ln|\cos x|-x\sin x \tan x-x \cos x+ \ln| \tan x +\sec x |+C$$

$$\int_{0}^{\pi}\dfrac{x}{1+\sin x}dx = \left [ x \tan x +\ln|\cos x|-x\sin x \tan x-x \cos x+ \ln| \tan x +\sec x | \right ]_0^{\pi}$$

$$\int_{0}^{\pi}\dfrac{x}{1+\sin x}dx = \pi$$
View Full Answer

Its FREE, you're just one step away


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

Realted Questions

Q1 Single Correct Medium
The integral $$\displaystyle \int^{4}_{2}\dfrac {\log x^{2}}{\log x^{2}+\log (36-12x+x^{2})}dx$$ is equal to
  • A. $$2$$
  • B. $$4$$
  • C. $$6$$
  • D. $$1$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q2 Subjective Medium
 Evaluate: $$\displaystyle\int _{ 0 }^{ 2 }{ x\sqrt { x+2 } dx } $$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q3 Subjective Medium
Write a value of 
$$\int { \tan ^{ 6 }{ x }  } \sec ^{ 2 }{ x } dx$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q4 Subjective Medium
Show that $$\displaystyle \int \left ( e^{\log x}+\sin x \right )\cos  dx=x\sin x+\cos x+\frac{1}{2}\sin^{2}x.$$

Asked in: Mathematics - Integrals


1 Verified Answer | Published on 17th 09, 2020

View Answer
Q5 Passage Medium
Consider two differentiable functions $$f(x), g(x)$$ satisfying $$\displaystyle 6\int f(x)g(x)dx=x^{6}+3x^{4}+3x^{2}+c$$ & $$\displaystyle 2 \int \frac {g(x)dx}{f(x)}=x^{2}+c$$. where $$\displaystyle f(x)>0    \forall  x \in  R$$

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives


1 Verified Answer | Published on 17th 08, 2020

View Answer