Mathematics

# Evaluate $\displaystyle\int_{0}^{\pi/4}(\sqrt{\tan}x+\sqrt{\cot}x)dx$

##### SOLUTION
Let $I=\displaystyle\int_{0}^{\pi/4}(\sqrt{\tan}x+\sqrt{\cot}x)dx$

$I=\displaystyle\int_{0}^{\pi/4}(\sqrt{\dfrac{\sin x}{\cos x}}+\sqrt{\dfrac{\cos x}{\sin x}})dx$

$=\displaystyle\int_0^{\pi/4}\dfrac{\sin x+\cos x}{\sqrt{\sin x \cos x}}dx$

$=\sqrt{2}\displaystyle\int_0^{\pi/4}\dfrac{\sin x+\cos x}{\sqrt{2\sin x \cos x}}dx$

$=\sqrt{2}\displaystyle\int_0^{\pi/4}\dfrac{\sin x+\cos x}{\sqrt{1-(\sin x -\cos x)^2}}dx$

Let $\sin x-\cos x=t.$ Then
$\Rightarrow$  $\cos x+\sin x dx =dt$            [ Differentiating both sides ]
When $x=0,$
$\sin 0-\cos 0=t\Rightarrow t=1$
When $x=\dfrac{\pi}{4},$
$\sin\dfrac{\pi}{4}-\cos\dfrac{\pi}{4}=t\Rightarrow$ t=0\thereforeI=\sqrt{2}\displaystyle\int_{-1}^0\dfrac{dt}{\sqrt{1-t^2}}=\sqrt{2}\left[\sin^{-1}t\right]_{-1}^0=\sqrt{2}\left[\sin^{-1}(0)-\sin^{-1}(-1)\right]=\dfrac{\pi}{\sqrt{2}}$Its FREE, you're just one step away Subjective Medium Published on 17th 09, 2020 Questions 203525 Subjects 9 Chapters 126 Enrolled Students 84 #### Realted Questions Q1 Single Correct Hard If$I_{1}=\displaystyle \int_{0}^{\pi/2}\cos(\sin x)dx$;$I_{2}=\displaystyle \int_{0}^{\pi/2}\sin(\cos x)dx$and$I_{3}=\displaystyle \int_{0}^{\pi/2}\cos xdx$, then • A.$I_{3}>I_{1}>I_{2}$• B.$I_{1}>I_{2}>I_{3}$• C.$I_{3}>I_{2}>I_{1}$• D.$I_{1}>I_{3}>I_{2}$Asked in: Mathematics - Integrals 1 Verified Answer | Published on 17th 09, 2020 Q2 Single Correct Medium A curve is represented parametrically by the equation$x=e^t \cos t$and$y=e^t \sin t$, where$t$is a parameter. Then, If$F(t)=\int (x+y)dt$, then the value of$F\left (\displaystyle \frac {\pi}{2}\right )-F(0)$is • A.$1$• B.$-1$• C.$0$• D.$e^{\tfrac{\pi}{2}}$Asked in: Mathematics - Integrals 1 Verified Answer | Published on 17th 09, 2020 Q3 Subjective Hard Solve$\displaystyle\int\limits_{\pi /4}^{\pi /2} {\dfrac{{\cos \theta }}{{{{\left[ {\cos \dfrac{\theta }{2} + \sin \dfrac{\theta }{2}} \right]}^3}}}d\theta } $Asked in: Mathematics - Integrals 1 Verified Answer | Published on 17th 09, 2020 Q4 Subjective Hard Find:$\int { \dfrac { { x }^{ 3 } }{ (1+{ x }^{ 2 })^{ 3 } }  } $Asked in: Mathematics - Integrals 1 Verified Answer | Published on 17th 09, 2020 Q5 Passage Medium Consider two differentiable functions$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

On the basis of above information, answer the following questions :

Asked in: Mathematics - Limits and Derivatives

1 Verified Answer | Published on 17th 08, 2020