Calculus - intergral problem!!?

Question

find the integral of:

e^(-3x)cos(5x)dx


thanks!!!

Answer 1

Ok, you have to do integration by parts twice. "Deep breath" and here we go.

Let u = cos(5x), dv = e^(-3x).
So, du = -5sin(5x), v = (-1/3)e^(-3x)

Using by parts we get,
(-1/3)e^(-3x)cos(5x) - (5/3)integral{e^(-3x)sin(5x)}

Use integration by parts to integrate e^(-3x)sin(5x)
Let u = sin(5x), dv = e^(-3x).
So, du = 5cos(5x), v = (-1/3)e^(-3x)

Using by parts we get,
(-1/3)e^(-3x)sin(5x) - (-5/3)integral{e^(-3x)cos(5x)}.

Putting it all together gives us that:
integral{e^(-3x)cos(5x)} =
(-1/3)e^(-3x)cos(5x) + (5/9)e^(-3x)sin(5x) - (25/9)integral{e^(-3x)cos(5x)}.

So, we can add (25/9)integral{e^(-3x)cos(5x)} to both sides and that would give us 34/9 times the integral, so multiply both sides by 9/34. Our answer is:
(-3/34)e^(-3x)cos(5x) + (5/34)e^(-3x)sin(5x) + C

Answer 2

One can solve this using integration by parts twice.
For example,
1st, take u=e^(-3x), dv=cos(5x)dx and integrate by parts
2nd, take u = e^(-3x), dv= sin(5x)dx and integrate by parts the remaining integral
At this point, one gets some x-terms and the initial integral back. Denote it by , say, A and solve for it .

Answer 3

??x-a million?dx from -2 to 5 this absolution function will become a piecewise function: ?x-a million dx for x ? a million and ?-(x-a million) dx fro x < a million so what we are gonna do is to locate the imperative of x-a million from a million to 5, then upload the imperative of -(x-a million) from -2 to a million ?-(x-a million) dx [-2,a million] + ?x-a million dx [a million,5] ?-x+a million) dx [-2,a million] + ?x+a million dx [a million,5] (-a million/2)x² + x [-2,a million] + (a million/2)x² - x [a million,5] evaluate: 9/2 + 8 25/2 so the respond is E wish it helps :-)

Answer 4

♠ thus y(x)=e^(-3x)cos(5x); since exp(jt) =cost +j*sint, then consider
z(x) = exp(-3x)*exp(5jx), where j^2=-1; and therefore
♣ Z(x) =∫dx*exp(-3x +5jx) = exp(-3x +5jx)/(-3+5j)=
= exp(-3x)*(cos(5x) +j*sin(5x))*(-3 -5j)/(9+25);
♦ now let us get rid of imaginary part we involved; thus
Y(x)= Re(Z(x)) =exp(-3x)*[-3cos(5x) +5sin(5x)]/34 +C;

Answer 5

e^(-3x)(5sin(5x)-3cos(5x))/34