Mathematics

# Evaluate the following integrals:$\displaystyle \int { \cfrac { 1 }{ { x }^{ 3 } } } \sin { \left( \log { x } \right) } dx\quad$

##### SOLUTION
Let $I=\displaystyle\int \dfrac{1}{x^3}\sin (\log x)d x$

Put $x=e^{t}\implies d x=e^t d t$

$\implies I=\displaystyle\int \dfrac{1}{(e^t)^3}\sin (\log e^t)\times e^t\ d t$

$\implies I=\displaystyle\int e^{-2 t}\sin t\ d t$

As we know that

$\displaystyle\int f(x) g (x) d x=f(x)\int g(x) d x-\int f'(x)\left(\int g(x) d x\right) d x$

Here $f(t)=\sin t,g(t)=e^{-2 t}$

$\implies f'(t)=\cos t$

$\implies \displaystyle\int g(t) d t=\int e^{-2 t} d t=\dfrac{e^{-2 t}}{-2}=-\dfrac{1}{2}e^{-2 t}$

$\implies I=\sin t\times \bigg(\dfrac{-1}{2}e^{-2 t}\bigg)-\displaystyle\int \cos t\times \bigg(\dfrac{-1}{2}e^{-2 t}\bigg)d t$

$\implies I=-\dfrac{1}{2}e^{-2 t}\sin t+\dfrac{1}{2}\displaystyle\int e^{-2 t}\cos t\ d t$

As we know that

$\displaystyle\int f(x) g (x) d x=f(x)\int g(x) d x-\int f'(x)\left(\int g(x) d x\right) d x$

Here $f(t)=\cos t,g(t)=e^{-2 t}$

$\implies f'(t)=-\sin t$

$\implies \displaystyle\int g(t) d t=\int e^{-2 t} d t=\dfrac{e^{-2 t}}{-2}=-\dfrac{1}{2}e^{-2 t}$

$\implies I=-\dfrac{1}{2}e^{-2 t}\sin t+\dfrac{1}{2}\left(\cos t\times \bigg(-\dfrac{1}{2}e^{-2 t}\bigg)-\displaystyle\int (-\sin t)\times \bigg(-\dfrac{1}{2}e^{-2 t}\bigg) d t\right)$

$\implies I=-\dfrac{1}{2}e^{-2 t}\sin t-\dfrac{1}{4}e^{-2 t}\cos t-\dfrac{1}{4}\displaystyle\int e^{-2 t}\sin t\ dt$

$\implies I=-\dfrac{1}{2}e^{-2 t}\sin t-\dfrac{1}{4}e^{-2 t}\cos t-\dfrac{1}{4}I$

$\implies I\bigg(1+\dfrac{1}{4}\bigg)=-\dfrac{1}{4}e^{-2 t}(2\sin t+\cos t)$

$\implies \dfrac{5}{4} I=-\dfrac{1}{4}e^{-2 t}(2\sin t+\cos t)$

$\implies I=-\dfrac{1}{5 (e^{t})^2}(2\sin t+\cos t)$

$\implies I=-\dfrac{1}{5 x^2}(2\sin (\log x)+\cos (\log x))$

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

#### Realted Questions

Q1 Single Correct Hard
$\displaystyle \int_{e^{e^{e}}}^{e^{e^{e^{e}}}}\frac{dx}{xlnx\cdot ln\left ( lnx \right )\cdot ln\left ( ln\left ( lnx \right ) \right )}$ equals
• A. 1/e
• B. e-1
• C. 1+e
• D. 1

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int_{-1 /2}^{1 /2} \displaystyle \cos x\log\left(\frac{1-x}{1+x}\right)dx$
• A. $\displaystyle \frac{1}{3}$
• B. $\displaystyle \frac{2}{3}$
• C. $\displaystyle \frac{2}{5}$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Prove that $\displaystyle\int^{\pi}_0\dfrac{x\sin x}{(1+\sin x)}dx=\pi\left(\dfrac{\pi}{2}-1\right)$.

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Solve:-
$\int\limits_0^{\dfrac{\pi}{2}} {\log \sin xdx}$

1 Verified Answer | Published on 17th 09, 2020

Q5 Single Correct Medium
Integrate $\dfrac1x$ with respect to $x:$
• A. $x \sin^2 x$
• B. $\sec^3 x$
• C. $2x^3 e^{x^2}$
• D. $\ell n x$