integrate e^xcosx dx. PLS HELP!!!!!!?

Question

will vote for quickest correct answers with appropriate workings

ok i've worked it out myself. 1/2(e^xsinx+e^xcosx) .
genius!

Answer 1

When u integrate anything with e^anything..the 'anything' remains the same but u need to do something else.
it would be - -1/xsinx + cosx . e^xcosx
try differentiating this and u can cancel it out to get the one in ur question..

Answer 2

First, enable's do a sprint algrebraic manipulation to make this much less complicated to combine. sin(x) + sin(2x) = sin(x) + 2sin(x)cos(x) = sin(x)[a million + 2cos(x)] Now enable's placed the fraction at the same time. a million/[sin(x) + sin(2x)] = a million/{sin(x)[a million + 2cos(x)]} = sin(x)/{sin²(x)[a million + 2cos(x)]} _____________ Now we are able to combine. critical[sin(x)/{sin²(x)[a million + 2cos(x)]}] dx u = cos(x) du = -sin(x)dx sin²(x) = a million - cos²(x) = a million - u² _________ = critical[a million/{(a million - u²)(a million + 2u)}] [-du] = critical[-a million/{(a million + u)(a million - u)(a million + 2u)}] du _________ Now enable's take a destroy from integration and decompose the expression into partial fractions. -a million/[(a million + u)(a million - u)(a million + 2u)] = A/(a million + u) + B(a million - u) + C(a million + 2u) Multiply via via (a million + u)(a million - u)(a million + 2u) to sparkling the denominators. -a million = A(a million - u)(a million + 2u) + B(a million + u)(a million + 2u) + C(a million + u)(a million - u) -a million = A(a million + u - 2u²) + B(a million + 3u + 2u²) + C(a million - u²) -a million = (-2A + 2B - C)u² + (A + 3B)u + (A + B + C) And we've 3 equations in 3 unknowns. 0 = -2A + 2B - C 0 = A + 3B -a million = A + B + C including all 3 equations we get: -a million = 6B B = -a million/6 From the 2nd equation we get: A = -3B = 3/6 = a million/2 From the third equation we get: C = -a million - A - B = -a million - a million/2 + a million/6 = -4/3 And we've: a million/[(a million + u)(a million - u)(a million + 2u)] = (a million/2)/(a million + u) - (a million/6)(a million - u) - (4/3)(a million + 2u) _____________ Now we are able to resume integrating. = critical[(a million/2)/(a million + u) - (a million/6)(a million - u) - (4/3)(a million + 2u)] du = (a million/2)ln|a million + u| + (a million/6)ln|a million - u| - (2/3)ln|a million + 2u| + C replace lower back. = (a million/2)ln|a million + cos(x)| + (a million/6)ln|a million - cos(x)| - (2/3)ln|a million + 2cos(x)| + C

Answer 3

Typing the answer is a tedious job. Any calculus book contains the answer. I would like to reference to any of them (some are online & free)