Integrate this monster

∫sec⁡3(x)1+sin⁡(x) dx= ? \large \int \sqrt{\frac{\sec^{3}(x)}{1+\sin(x)}} \, dx =\ ?

#Calculus #Integrals

Note by Majed Musleh
6 years ago

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Comments

It's equals to ∫1cos⁡3(x)(1+sin⁡(x)) dx \displaystyle \int \sqrt{\frac1{\cos^3(x) (1+\sin(x))}} \, dx . Let y=π2−xy = \frac \pi2 - x , then it becomes

−∫1sin⁡3(y)(1+cos⁡(y)) dy -\int \frac1{\sqrt{\sin^3(y)(1+\cos(y))}} \, dy

Apply Tangent half-angle substitution, then it equals to

−∫1(2t1+t2)3(1+1−t21+t2)⋅2dtt2+1=−12∫1+t2t3/2 dt - \int \sqrt{\frac{1}{\left(\frac{2t}{1+t^2}\right)^3\left(1+ \frac{1-t^2}{1+t^2}\right)} }\cdot \frac{2 dt}{t^2+1} = -\frac12 \int \frac{1+t^2}{t^{3/2}} \, dt

Which can be easily integrated from here, back substitute everything and you're done.

Pi Han Goh - 6 years ago
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