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Mathematics Question on Integrals of Some Particular Functions

Evaluate ∫π/43π/411+cos xdx\int\limits^{3\pi/4}_{\pi/4} \frac{1}{1+cos\,x}dx

A

22

B

−2-2

C

1/21/2

D

−1/2-1/2

Answer

22

Explanation

Solution

The correct answer is A:2
∫π43π4dx1+cosx\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{dx}{1+cosx}
=∫π43π41−cosx(1−cosx)(1+cosx)dx=\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{1-cosx}{(1-cosx)(1+cosx)}dx
=∫π43π41−cosx1−cos2xdx=\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{1-cosx}{1-cos^2x}dx
=∫π43π41−cosxsin2xdx=∫π43π41sin2x−∫π43π4cosxsin2xdx=\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{1-cosx}{sin^2x}dx=\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{1}{sin^2x}-\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}\frac{cosx}{sin^2x}dx
=∫π43π4cosec2xdx−∫π43π4cotx.cosecxdx=\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}cosec^2xdx-\int^{\frac{3\pi}{4}}_{\frac{\pi}{4}}cotx.cosecxdx
=−cot(3π4)+cosec(3π4)−(cot(π4)−cosec(π4))=-cot(\frac{3\pi}{4})+cosec(\frac{3\pi}{4})-(cot(\frac{\pi}{4})-cosec(\frac{\pi}{4}))
=2=2
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