Question

using integration by parts what is the answer of this integral?

Answer 1

this is the answer i got....

Answer 2

your answer looks about right!

Answer 3

u = e^x and dv = cosx
du = e^x dx and v = sinx

\[∫u dv = uv - ∫v du\]
\[= e^x sinx - ∫e^x sinx dx\]

integrate by parts again
\[u = e^x~ and ~dv = sinx\]
\[du = e^x ~dx~ and~ v ~= ~-cosx\]

\[= e^x sinx - [-e^x cosx - ∫-e^x cosx dx]\]
\[= e^x sinx - [-e^x cosx + ∫e^x cosx dx]\]
\[= e^x sinx + e^x cosx - ∫e^x cosx dx\]

rewriting
\[∫e^x cosx dx = e^x sinx + e^x cosx - ∫e^x cosx dx\]

so you'll have
\[2∫e^x cosx = e^x sinx + e^x cosx\]
\[∫e^x cosx = \frac{1}{2}(e^x sinx + e^x cosx) + C \]