Mathematics

# $\displaystyle \int_{\pi /5}^{3\pi /10}\frac{\cos x}{\cos x+\sin x}dx$is equal to

none of these

##### SOLUTION
Let $\displaystyle I=\int _{ \frac { \pi }{ 5 } }^{ \frac { 3\pi }{ 10 } }{ \frac { \cos { x } }{ \cos { x } +\sin { x } } } dx$

Multiply numerator and denominator by $sec^{ 3 }{ \left( x \right) }$
$\displaystyle I=\int _{ \frac { \pi }{ 5 } }^{ \frac { 3\pi }{ 10 } }{ \frac { sec^{ 2 }{ x } }{ sec^{ 2 }{ x }+sec^{ 2 }{ x }\tan { x } } } dx=\int _{ \frac { \pi }{ 5 } }^{ \frac { 3\pi }{ 10 } }{ \frac { sec^{ 2 }{ x } }{ 1+\tan { x } +\tan ^{ 2 }{ x } +\tan ^{ 3 }{ x } } } dx$

Put $t=\tan { x } \Rightarrow dt=sec^{ 2 }{ x }dx$
$\displaystyle \therefore I=\int _{ \tan { \frac { \pi }{ 5 } } }^{ \tan { \frac { 3\pi }{ 10 } } }{ \frac { 1 }{ { u }^{ 2 }+{ u }^{ 3 }+u+1 } } du$

$\displaystyle=\int _{ \tan { \frac { \pi }{ 5 } } }^{ \tan \frac { 3\pi }{ 10 } }{ \left( \frac { 1-u }{ 2\left( { u }^{ 2 }+1 \right) } +\frac { 1 }{ 2\left( u+1 \right) } \right) du }$

$\displaystyle =\frac { 1 }{ 2 } \int _{ \tan { \frac { \pi }{ 5 } } }^{ \tan { \frac { 3\pi }{ 10 } } }{ \left( \frac { 1 }{ { u }^{ 2 }+1 } -\frac { u }{ { u }^{ 2 }+1 } \right) } du+\frac { 1 }{ 2 } \int _{ \tan { \frac { \pi }{ 5 } } }^{ \tan { \frac { 3\pi }{ 10 } } }{\dfrac{1} {u+1} } du$

$\displaystyle =\left[ -\frac { 1 }{ 4 } \log { \left( { u }^{ 2 }+1 \right) +\frac { 1 }{ 2 } } \log { \left( u+1 \right) +\frac { 1 }{ 2 } \tan ^{ -1 }{ u } } \right] _{ \log { \frac { \pi }{ 5 } } }^{ \log { \frac { 3\pi }{ 10 } } }=\frac { \pi }{ 20 }$

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Single Correct Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

#### Realted Questions

Q1 Single Correct Medium
$\displaystyle \int (e^\sqrt[3]{x}dx)=$
• A. $x^{2/3}-2 x^{1/3} +2+c$
• B. $(x^{2/3} - 2x^{1/3} +2)\exp(\sqrt[3]{x})+c$
• C. $2(x^{2/3} - 2x^{1/3}+2)\exp(\sqrt[3]{x})+c$
• D. $3(x^{2/3} - 2x^{1/3}+2)\exp(\sqrt[3]{x})+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Solve
$\int (4x^{3}-\dfrac{3}{x^{4}}) dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Assertion & Reason Hard
##### ASSERTION

If $n>1$ then Statement -1: $\displaystyle \int_{0}^{\infty}\frac{dx}{1+x^{n}}=\int_{0}^{1}\frac{dx}{(1-x^{n})^{1/n}}$

##### REASON

Statement -2: $\displaystyle \int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Assertion is correct but Reason is incorrect
• C. Both Assertion and Reason are incorrect
• D. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
Suppose for every integer $n, . \displaystyle \int_{n}^{n+1} f(x)dx=n^{2}$ The value of $\displaystyle \int_{-2}^{4} f(x)dx$ is :
• A. $16$
• B. $14$
• C. $21$
• D. $19$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\int\dfrac{1}{\sec x+\tan x +\csc x +\cot x }dx=$
• A. $\dfrac{1}{2}(\sin x+ cos x+x)+C$
• B. $\dfrac{1}{2}(\sin x+ cos x-x)+C$
• C. $\dfrac{1}{2}(-\sin x+ cos x-x)+C$
• D. $\dfrac{1}{2}(\sin x- cos x-x)+C$