Mathematics

# Evaluate $\int \dfrac{\log (1+x)}{(1+x)}dx$

##### SOLUTION
$\begin{array}{l} \int { \dfrac { { \log \left( { 1+x } \right) } }{ { 1+x } } dx } \\ u=\log \left( { 1+x } \right) \, \, \, \, du=\dfrac { 1 }{ { 1+x } } dx \\ \int { u\, \, du=\dfrac { { { u^{ 2 } } } }{ 2 } +c } \\ =\dfrac { 1 }{ 2 } { \left( { \log \left( { 1+x } \right) } \right) ^{ 2 } }+c \end{array}$

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Subjective Hard Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 105

#### Realted Questions

Q1 Single Correct Hard
By Trapezoidal rule, the value of $\displaystyle\int _{ 0 }^{ 1 }{ { x }^{ 3 }dx }$, considering five sub-intervals, is
• A. $0.20$
• B. $0.28$
• C. $0.27$
• D. $0.26$

1 Verified Answer | Published on 17th 09, 2020

Q2 Multiple Correct Hard
If $I_n = \displaystyle \int_{-\pi}^{\pi} \dfrac{sinnx}{(1+\pi^x)sinx}dx,$ n=0,1,2,..., then
• A. $I_n=I_{n+1}$
• B. $I_n=I_{n+2}$
• C. $\displaystyle \sum_{m=1}^{10} I_{2m+1}=10\pi$
• D. $\displaystyle \sum_{m=1}^{10} I_{2m}=0$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
Evaluate : $\displaystyle \int \frac { cot x}{( cosec x - cot x)} dx$
• A. $cosec x - cot x -x +C$
• B. $-cosec x +cot x -x + C$
• C. $-sin x + x+ C$
• D. $-cosec x -cot x- x +C$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate: $\displaystyle\int {\dfrac{{{x^2}}}{{1 + {x^6}}}\;dx}$

Let $n \space\epsilon \space N$ & the A.M., G.M., H.M. & the root mean square of $n$ numbers $2n+1, 2n+2, ...,$ up to $n^{th}$ number are $A_{n}$, $G_{n}$, $H_{n}$ and $R_{n}$ respectively.