Mathematics

$\int _{ 0 }^{ \pi /4 }{ x.\sec ^{ 2 }{ x } dx=? }$

$\frac {\pi}{4}+\log {\sqrt {2}}$

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

Realted Questions

Q1 Single Correct Hard
$\displaystyle \int { \sqrt { x } { e }^{ \sqrt { x } } } dx$ is equal to:
• A. $\displaystyle 2\sqrt { x } -{ e }^{ \sqrt { x } }-4\sqrt { { xe }^{ \sqrt { x } } } +c$
• B. $\displaystyle \left( 2x+4\sqrt { x } +4 \right) { e }^{ \sqrt { x } }+c$
• C. $\displaystyle \left( 1-4\sqrt { x } \right) { e }^{ \sqrt { x } }+c$
• D. $\displaystyle \left( 2x-4\sqrt { x } +4 \right) { e }^{ \sqrt { x } }+c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
The function
$f(x) = \int_1^x [ 2 (t-1) (t-2)^3+3(t-1)^2 (t-2)^2] dt$ has :
• A. Minimum at $x = \dfrac {7}{5}$
• B. Neither maximum nor minimum at $x=2$
• C. All of these
• D. Maximum at $x=1$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Hard
Let $f\left(x\right)$ be an even function,${I}_{1}=\displaystyle\int_{0}^{\frac{\pi}{2}}{f\left(\cos{2x}\right)\cos{x}dx}$ and ${I}_{2}=\displaystyle\int_{0}^{\frac{\pi}{4}}{f\left(\sin{2x}\right)\cos{x}dx}$ then $\dfrac{{I}_{1}}{{I}_{2}}=$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Find the following integrals
$\displaystyle\int \cfrac{x^{3}-x^{2}+x-1}{x-1} d x$

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