Question

integration of sinx×cosx

Answer 1

First let's use a substitution to break it down a bit:

w = cos(x)
dw = -sin(x) * dx

Substitute accordingly:

∫ -ln(w) * dw

Now use integration by parts:

u = ln(w)
dw = 1/w

dv = -1
v = -w

Apply the integration by parts formula:

uv - ∫ vdu

-wln(w) - ∫ -1 * dw
-wln(w) + ∫ 1 * dw

Integrate:

= -wln(w) + w + C

Now substitute cos(x) back for w:

= -cos(x) * ln[cos(x)] + cos(x) + C