Mathematics

# $\int\limits_0^\pi {\frac{{{x^2}{{\cos }^4}x\sin x}}{{2\pi x - {\pi ^2}}}dx} =$

$\frac{\pi }{5}$

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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
The value of $\displaystyle \int_0^{\cfrac {\pi}{2}}\log \left (\frac {4+3 \sin x}{4+3 \cos x}\right )dx$ is
• A. $2$
• B. $\frac {3}{4}$
• C. $-2$
• D. $0$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
The value of the definite integral $\displaystyle \int_{0}^{\pi / 2} \dfrac{\sin 5 x}{\sin x} d x$ is
• A.
• B. $\pi$
• C. $2 \pi$
• D. $\dfrac{\pi}{2}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Integrate by using suitable substitution
(1). $\int {\sqrt {3 - 2s} }$ ds
(2). $\int {\csc \left( {{{v - \pi } \over 2}} \right)} \cot \left( {{{v - \pi } \over 2}} \right)dv$
(3). $\int\limits_\pi ^{2\pi } {\theta d\theta }$
(4). $\int\limits_0^{\sqrt x } {x\sin {x^2}dx}$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
The value of the integral $\underset{0}{\overset{\pi / 4}{\int}} \dfrac{\sin \, x + \cos \, x}{3 + \sin \, 2x} dx$, is
• A. $\log 2$
• B. $\log 3$
• C. $\dfrac{1}{8} \log \, 3$
• D. $\dfrac{1}{4} \log \, 3$

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 $\displaystyle f(x)>0 \forall x \in R$