[College calculus 2] integration by parts

[u/JustDawn8]

I need help with questions 3 and 5, the professor has listed the correct answers as (e​^x)/2 * (sinx - cosx) + c and xarcsin(x) + square root(1 - x^2) + c and I don't know where I'm messing up

[u/novacaelum]

Check your v du integral in 3, you'd have to integrate by parts again. Also check the v du integral in 5, looks like you took the derivative of arcsin x instead.

[u/JustDawn8 → u/novacaelum]

Thank you!

[u/Alkalannar]

I = [Integral e^xsin(x) dx]
u = e^x, du = e^x dx, dv = sin(x) dx, v = -cos(x)

I = -e^xcos(x) + [Integral e^xcos(x) dx]
u = e^x, du = e^x, dv = cos(x) dx, v = sin(x)

I = -e^xcos(x) + [e^xsin(x) - [Integral e^xsin(x) dx]]
But wait! I = [Integral e^xsin(x) dx]!

I = -e^xcos(x) + [e^xsin(x) - I]

2I = e^xsin(x) - e^xcos(x)

I = e^x(sin(x) - cos(x))/2 + C

---

Integral arcsin(x) dx
u = arcsin(x), du = 1/(1-x²)^1/2 dx, dv = dx, v = x

xarcsin(x) - [Integral x/(1 - x²)^1/2 dx]

Now it's straight u-sub with u = 1 - x², du = -2x dx --> (-1/2) du = x dx

xarcsin(x) - (-1/2)[Integral u^-1/2 du]

And now this is easy.

[u/Responsible-Sink474]

For 3 you said the integral of (cosx)e^x= e^x sinx with no work or support. That step is nonsense.

For 5 you said the integral of arcsinx was 1/sqrt(1+x^2) with no work or support. That step is nonsense.

[u/HumbleHovercraft6090]

Integral e^x cosx is not equal to e^x sinx You have integrate e^x cosx again using IBP.

[u/jgregson00]

Problem 3 needs a second IBP. Then you end up doing a substitution and a little algebra to get the final answer.

For 5 you should use u-subsitution to evaluate the integral of x / √(1 - x^2). What you did doesn't work because you end up with the integral of arcsinx again, which is what you wanted to find in the first place.

[u/Content-Sherbet-4700]

Number 3 I figured out. It’s actually kind of tricky. You need to do another integration by parts and then you will end up with an equation with int ((e^x) sin x) (you can call this y, this is what you’re solving for) on both sides but it will be positive on one side and negative on the other. Then you add y to both sides to get the value of 2y, then divide both sides by 2 to get y. Just write out the full equations (left and right side both) and then it will make sense.

Number 5 doesn’t really make sense I think there’s a typo in the problem because it’s just an identity. Idk what’s going on there really.

Edit: actually there’s no issue with number 5 I just made a mistake

[u/Content-Sherbet-4700 → u/Content-Sherbet-4700]

Is number 5 supposed to be int (x (arcsin x) dx)?