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Grade 12Integral Calculus

Integrate 1/(1+(tanx)^(1/3)) from π/6 to π/3................

Mihir Gupta

6 Years agoGrade 12

2 Answers

Arun

6 Years ago

Dear Mihir

I have atta hed the image containing solution. If you find any difficulty, please feel free to ask again.

$I = \frac{\pi }{12}$ $2I = \frac{\pi }{6}$ $2I = [x]_{\pi /6}^{\pi /3}$ $2I =\int_{\pi /6}^{\pi /3}dx$ $2I =\int_{\pi /6}^{\pi /3}\frac{\sqrt{cosx}+\sqrt{sin(x)}}{\sqrt{sin(x)}+\sqrt{cos(x)}}dx$ ......(2)(1) + (2) $I =\int_{\pi /6}^{\pi /3}\frac{\sqrt{sin(x)}}{\sqrt{sin(x)}+\sqrt{cos(x)}}dx$ $I =\int_{\pi /6}^{\pi /3}\frac{\sqrt{cos(\frac{\pi }{2}-x)}}{\sqrt{cos(\frac{\pi }{2}-x)}+\sqrt{sin(\frac{\pi }{2}-x)}}dx$ $\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$ …......(1) $I =\int_{\pi /6}^{\pi /3}\frac{\sqrt{cosx}}{\sqrt{cosx}+\sqrt{sinx}}dx$ $I =\int_{\pi /6}^{\pi /3}\frac{1}{1+\sqrt{tanx}}dx$

Aditya Gupta

6 Years ago

actually aruns ans is wrong coz the power in ques is 1/3

let J= ∫1/(1+(tanx)^(1/3))dx from π/6 to π/3.......(1)

now we know the property that f(x) can be replaced by f(a+b – x). here a+b= π/6 + π/3= pi/2.

also, tan(pi/2 – x)= cotx

so J= ∫1/(1+(cotx)^(1/3))dx from π/6 to π/3

or J= ∫(tanx)^(1/3)/(1+(tanx)^(1/3))dx from π/6 to π/3........(2)

adding 1 and 2

2J= ∫(1+(tanx)^(1/3))/(1+(tanx)^(1/3))dx from π/6 to π/3.

2J= ∫dx from π/6 to π/3.

2J= pi/6

or J= π/12

kindly approve :)

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