integrate: (0 to pi/4) 1/cos^3x (whole root)(2sin 2x): - Maths - Integrals

Question

integrate: (0 to pi/4) 1/cos^3x (whole root)(2sin 2x):

Ishwarmani · Accepted Answer

Dear
Student,
Please find
below the solution to the asked query:
I=∫0π41cos3x2sin2xdxDividing the numerator and denominator by cos4x, we getI=∫0π4sec4x2sin2xcosxdx=∫0π4sec2x sec2x4sinx cosxcos2xdx=12∫0π41+tan2xsec2xtanxdxLet tanx=t, sec2xdx=dtWhen x→0, then t→0 andwhen x→π4, then t→1So, I=12∫011+t2dtt=12∫011t+t2tdt=12∫01t-12dt+∫01t32dt=12t-12+1-12+1+t32+132+101=122t+2t52501=122+25-0+0=12125∵I=65
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