A Maths C (Advanced Maths Integral Question)?

First consider the first term of the integrand.

∫ sinx(cos²(x)) dx

You can evaluate this integral by letting u=cosx

then

du = -sinx dx

If you multiply and divide by -1 and then substitute these values into the integral , you have

-∫ du/u²

which is the same as

-∫ u^-2 du

= u^-1 + C

= 1/cosx + C = secx + C

The other two terms are pretty straight forward. I come up with

∫ [ sin x/cos2 x - 5cos x+4 ] dx

= sec(x) - 5sin(x) + 4x + C

This is a simple antiderivative integral.

Since the derivative of sinx is cosx and sinx/(cosx)^2=secxtanx

Int { secxtanx -5cosx + 4 }

= secx -5sinx +4x + C ANSWER

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