Separable differential equation and Integration by parts

[BarackObama]

Homework Statement

dy/dx = e^ysin^2x/ysecx

Stewart 6e 10.3 # 8

Homework Equations

The Attempt at a Solution

ydy/e^y = sin^2xdx/secx
e^-ydy = sec^-1xsin^2xdx

Integration by parts
u = e^-y
du = -e^-y
dv = ydy
v = y^2/2

∫udv = e^-yy^2/2 + ∫y^2/2e^-y
= y^2/2e^y + y^3/6?
another integration by parts?

 

[snipez90]

If you have something of the form (y^m)e^(ny) which you need to integrate by parts, you want to differentiate y^m and integrate e^(ny), because this reduces the exponent on the y in subsequent integrals. Note you concluded with "another integration by parts" because you made the problem more difficult.

Also, be flexible. While this is easy to integrate by parts, note that differentiating y*e^-y is going to get you a y*e^-y term back (except maybe with a minus in front). In fact, letting f(y) = -y*e^-y, we see that f'(y) = y*e^-y - e^-y, so that adding Ce^-y to f(y) originally would have given you back y*e^-y upon differentiating (for an appropriate C, whose value should be obvious). Then f(y) + Ce^-y would be your antiderivative.

 

[BarackObama]

I'm still not sure how to proceed with this one... how do I integrate the right hand side?

 

[Char. Limit]

I'm still not sure how to proceed with this one... how do I integrate the right hand side?

1/sec(x) = cos(x)

 

[BarackObama]

Ok, I can do that substitution and then integrate by parts.

Still not sure how to integrate the left side though.

 

[Char. Limit]

Ok, I can do that substitution and then integrate by parts.

Still not sure how to integrate the left side though.

Why on Earth would you integrate by parts? That particular integral is best handled with a u-substitution.

 

[BarackObama]

right, let u = sin^2x

can you show me how to integrate the left side?

 

[snipez90]

Right hand side: Let u = sin(x), not sin^2(x)

Left hand side: Let u = y, dv = e^-y dy