Question

I'm really having trouble with this current topic that we're learning. Any explanations are greatly appreciated.

Evaluate the integrals:

1.) ∫ (2-2cos^2x) dx

2.) ∫ cot3x dx

3.) ∫ ((e^(sqrt x) / (sqrt x) dx))

Answer 1

1.) ∫ (2-2cos^2x) dx
= ∫ (2-2cos^2x) dx
= ∫ (1-(2cos^2x-1)) dx
= ∫ (1-cos 2x) dx
= x - (1/2)sin 2x + c

2.
∫ cot3x dx
= ∫ cos3x/sin 3x dx
= (1/3) ln(sin 3x) + c

3.
∫ (e^√x / √x dx
= ∫ e^(x^(1/2)) (e^(-1/2)) dx
= 2 e^(x^(1/2)) + c or 2 e^√x + c