Mathematics

# Find $\displaystyle \int_0^{1/4 \pi} ln(1 + \tan x )dx$.

##### SOLUTION
$I = \displaystyle \int_0^{\dfrac{\pi}{4}} \ln (1 + \tan x) dx$
Use king
$I = \displaystyle\int_0^{\dfrac{\pi}{4}} \ln \left(1 + \tan \left(\dfrac{\pi}{4} - 4\right)\right) dx$
$= \displaystyle\int_0^{\dfrac{\pi}{4}} \ln \left(1 + \dfrac{1 - \tan x}{1 + \tan x} \right) dx$
$= \displaystyle\int_0^{\dfrac{\pi}{4}} \ln \left(\dfrac{1 + \tan x + 1 - \tan x}{1 + \tan x } \right) dx$
$= \displaystyle\int_0^{\dfrac{\pi}{4}} \ln \left(\dfrac{2}{1 + \tan x} \right) dx$
$2 I = \displaystyle\int_0^{\dfrac{\pi}{4}} \ln \left(\dfrac{2}{1 + \tan x} \right) + \ln (1 + \tan x ) dx$
$2 I = \displaystyle\int_0^{\dfrac{\pi}{4}} \ln (2) dx$
$2I = \ln (2) \left(\dfrac{\pi}{4} - 0\right)$
$I = \dfrac{\pi}{8} \ln (2)$

Its FREE, you're just one step away

Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Medium
Evaluate $\displaystyle \int { \frac { \cos { x } +x\sin { x } }{ { x }^{ 2 }+\cos ^{ 2 }{ x } } dx }$
• A. $\ln\left(\dfrac { x+\cos { x } }{ x } \right)+c$
• B. $\tan^{-1}\left(\dfrac { x }{ x+\cos { x } } \right)+c$
• C. None
• D. $-\tan^{-1}\left(\dfrac { \cos { x } }{ x } \right)+c$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Evaluate $\displaystyle\int{\frac{dx}{\sqrt{(x-a)(b-x)}}}$.
• A. $\displaystyle I=2\cos^{-1}{\sqrt{\frac{x-a}{(b-a)}}}+C$
• B. $\displaystyle I=\sin^{-1}{\sqrt{\frac{x-a}{(b-a)}}}+C$
• C. $\displaystyle I=2\sin^{-1}{\sqrt{\frac{x-b}{(a-b)}}}+C$
• D. $\displaystyle I=2\sin^{-1}{\sqrt{\frac{x-a}{(b-a)}}}+C$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Assertion :$\displaystyle \int\cfrac{3x+7}{3x^{2}+14x-5}dx= \cfrac{1}{2}\log |3x^{2}+14x-5|+c$

Reason :$\displaystyle \int\cfrac{f'(x)}{f(x)}dx=\log|f(x)|+c$
• A. Both Assertion and Reason are true but Reason is not correct explanation of Assertion
• B. Assertion is true but Reason  is false
• C. Assertion is false but Reason is true
• D. Both Assertion and Reason are true and Reason is the correct explanation of Assertion

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q4 Assertion & Reason Hard
##### ASSERTION

If $a>0$ and $b^2-4ac<0$, then the value of the integral $\displaystyle\int{\frac{dx}{ax^2+bx+c}}$ will be of the type $\displaystyle\mu\tan^{-1}{\frac{x+A}{B}}+C$, where $A,B,C,\mu$ are constants.

##### REASON

If $a>0$, $b^2-4ac<0$, then $ax^2+bx+C$ can be written as sum of two squares.

• A. Both Assertion and Reason are correct but Reason is not 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 and Reason is the correct explanation for Assertion

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
$\int\limits_0^\pi {\log (1 + cosx)dx = }$
• A. $- \frac{\pi }{2}\log 2$
• B. $- \frac{\pi }{3}$
• C. $- 2\pi \log 2$
• D. $- \pi \log 2$

Asked in: Mathematics - Integrals

1 Verified Answer | Published on 17th 09, 2020