Mathematics

# Evaluate: $\int\limits_\pi ^{\tfrac{{5\pi }}{4}} {\dfrac{{\sin2x.dx}}{{\cos {^4}x + \sin {^4}x}} }$

$\dfrac{\pi }{4}$

##### SOLUTION
$\int _{ \pi }^{ \dfrac { 5\pi }{ 4 } }{ \dfrac { \sin { 2x } dx }{ \cos ^{ 4 }{ x } +\sin ^{ 4 }{ x } } } =\int _{ \pi }^{ \dfrac { 5\pi }{ 4 } }{ \dfrac { \sin { 2x } dx }{ { \left( \cos ^{ 2 }{ x } +\sin ^{ 2 }{ x } \right) }^{ 2 }-2\sin ^{ 2 }{ x } \cos ^{ 2 }{ x } } }$

$=\int _{ \pi }^{ \dfrac { 5\pi }{ 4 } }{ \cfrac { \sin { 2x } dx }{ { 1-\dfrac { 1 }{ 2 } \left( \sin ^{ 2 }{ 2x } \right) } } } =\int _{ \pi }^{ \dfrac { 5\pi }{ 4 } }{ \cfrac { \sin { 2x } dx }{ 1+1-\sin ^{ 2 }{ 2x } } } =\int _{ \pi }^{ \dfrac { 5\pi }{ 4 } }{ \cfrac { \sin { 2x } }{ 1+\cos ^{ 2 }{ 2x } } } dx$

$\cos { 2x } =t\Rightarrow -2\sin { 2x } dx=dt$

$\int _{ \pi }^{ \dfrac { 5\pi }{ 4 } }{ \dfrac { -1 }{ 1+{ t }^{ 2 } } } dt={ \left[ -\tan ^{ -1 }{ t } \right] }_{ \pi }^{ \dfrac { 5\pi }{ 4 } }={ \left[ -\tan ^{ -1 }{ \cos { 2x } } \right] }_{ \pi }^{ \dfrac { 5\pi }{ 4 } }=0+\dfrac { \pi }{ 4 } =\dfrac { \pi }{ 4 }$

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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
If $\displaystyle{\int \frac{\displaystyle dx}{\displaystyle \sqrt{x}+\displaystyle \sqrt[3]{x}}}=a\sqrt{x}+b(\sqrt[3]{x})+c(\sqrt[6]{x})+d\: \ln(\sqrt[6]{x}+1)+e$, $e$ being arbitrary constant then. Find the value of $20a + b + c + d.$
• A. $35$
• B. $43$
• C. $47$
• D. $37$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Medium
$\displaystyle \int{(x^{3}a+x^{2}a+xa)(2x^{2}a+3xa+6)^{1/a}dx}$=
• A. $\dfrac{1}{6(a+1)}(2x^{3a}+3x^{2a}+6x^{a})^{1+\dfrac{1}{a}}+C$
• B. $\dfrac{1}{3(a+1)}(2x^{3a}+3x^{2a}+6x^{a})^{1+\dfrac{1}{a}}+C$
• C. $none\ of\ these$
• D. $\dfrac{1}{6(a+1)}(2x^{3a}+3x^{2a}+6x^{a})^{1-\dfrac{1}{a}}+C$

1 Verified Answer | Published on 17th 09, 2020

Q3 Multiple Correct Hard
$\displaystyle \int_{0}^{\pi/2}\frac{dx}{1+\tan x}$
• A. a multiple of $\pi/2$
• B. a multiple of $\pi$
• C. a multiple of $\pi/4$
• D. equal to $\pi/4$

1 Verified Answer | Published on 17th 09, 2020

Q4 Subjective Medium
Evaluate the following integral:
$\displaystyle\int^2_0x\sqrt{2-x}dx$.

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