Mathematics

# Solve :$\displaystyle \int_0^{\pi/2}\dfrac{dx}{1+\tan^3 x}$ ?

$\dfrac{\pi}{4}$

##### SOLUTION
$\quad \quad \int _{ 0 }^{ \dfrac { \pi }{ 2 } }{ \dfrac { dx }{ 1+{ \tan }^{ 3 }x } } \\ =\int _{ 0 }^{ \dfrac { \pi }{ 2 } }{ \dfrac { 1 }{ \left( 1+{ u }^{ 2 } \right) \left( 1+{ u }^{ 3 } \right) } du } \quad \quad \left[ let,\quad u=\tan x \right] \\ =\int _{ 0 }^{ \dfrac { \pi }{ 2 } }{ \dfrac { 1-2u }{ 3\left( { u }^{ 2 }-u+1 \right) } } +\dfrac { u+1 }{ 2\left( { u }^{ 2 }+1 \right) } +\dfrac { 1 }{ 6\left( u+1 \right) } du\\ =\int _{ 0 }^{ \dfrac { \pi }{ 2 } }{ \dfrac { 1-2u }{ 3\left( { u }^{ 2 }-u+1 \right) } du } +\int _{ 0 }^{ \dfrac { \pi }{ 2 } }{ \dfrac { u+1 }{ 2\left( { u }^{ 2 }+1 \right) } du } +\int _{ 0 }^{ \dfrac { \pi }{ 2 } }{ \dfrac { 1 }{ 6\left( u+1 \right) } du } \\ ={ \left[ -\dfrac { 1 }{ 3 } log\left[ { u }^{ 2 }-u+1 \right] \right] }_{ 0 }^{ \dfrac { \pi }{ 2 } }+{ \left[ \frac { 1 }{ 2 } \left( \dfrac { 1 }{ 2 } log\left( { u }^{ 2 }+1 \right) +\tan ^{ -1 }{ \left( u \right) } \right) \right] }_{ 0 }^{ \dfrac { \pi }{ 2 } }+{ \left[ \dfrac { 1 }{ 6 } log\left( u+1 \right) \right] }_{ \left( 0 \right) }^{ \dfrac { \pi }{ 2 } }\\ =\dfrac { \pi }{ 4 } -0\\ =\dfrac { \pi }{ 4 }$

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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
$\int { \cfrac { dx }{ \sqrt { { x }^{ 2 }-6x+10 } } } =$?
• A. $\log { \left| x+\sqrt { { x }^{ 2 }-6x+10 } \right| } +C$
• B. $\log { \left| x-\sqrt { { x }^{ 2 }-6x+10 } \right| } +C$
• C. none of these
• D. $\log { \left| \left( x-3 \right) +\sqrt { { x }^{ 2 }-6x+10 } \right| } +C$

1 Verified Answer | Published on 17th 09, 2020

Q2 Single Correct Hard
Value of $\displaystyle \int_{a}^{\infty }\displaystyle \frac{dx}{x^{4}\sqrt{a^{2}+x^{2}}}$ is
• A. $\displaystyle \frac{2+\sqrt{2}}{3a^{4}}$
• B. $\displaystyle \frac{2-\sqrt{2}}{3a^{2}}$
• C. $\displaystyle \frac{\sqrt{2}+1}{3a^{2}}$
• D. $\displaystyle \frac{2-\sqrt{2}}{3a^{4}}$

1 Verified Answer | Published on 17th 09, 2020

Q3 Subjective Medium
Solve :
$\displaystyle\int { \cfrac { x }{ \sqrt { x+4 } } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Hard
$\displaystyle \int \frac{1}{\left ( x+1 \right )\sqrt{x^{2}+x+1}}dx$
• A. $\displaystyle =-\log \left | \frac{1}{x+1}-\frac{1}{2}+\frac{\sqrt{x^{2}+x+1}}{x-1} \right |$
• B. $\displaystyle =-\log \left | \frac{1}{x-1}-\frac{1}{2}+\frac{\sqrt{x^{2}+x-1}}{x-1} \right |+C$
• C. $\displaystyle =-\log \left | \frac{1}{x-1}-\frac{1}{2}+\frac{\sqrt{x^{2}+x+1}}{x+1} \right |+C$
• D. $\displaystyle =-\log \left | \frac{1}{x+1}-\frac{1}{2}+\frac{\sqrt{x^{2}+x+1}}{x+1} \right |+C$

1 Verified Answer | Published on 17th 09, 2020

Q5 Passage Medium
Let $F: R\rightarrow R$ be a thrice differential function. Suppose that $F(1) = 0, F(3) = -4$ and $F'(x)<0$ for all $x\in\left(\dfrac{1}{2},3\right)$. Let $f(x) = xF(x)$ for all $x\in R$.