How do I fix my definite integral issue?

[tangibleLime]

Homework Statement

Integrate from 0 to sqrt(pi/2). I can't figure out how to make the LaTeX work correctly for it, so it just looks like an indefinite integral.

$$\int 7\theta^3cos(\theta^2) d\theta$$​

Homework Equations

$$\int udv = uv - \int vdu$$​

The Attempt at a Solution

First I took the 7 out of the integral and pulled a theta out of theta cubed in order to match it with the theta squared.

$$7 \int \theta^2cos(\theta^2)\theta d\theta$$​

Then I substituted $$w = \theta^2, dw = 2\theta d\theta, \frac{dw}{2}=\theta d\theta$$.

To make the bounds of integration to work with this substitution...

$$w = 0^2 = 0$$
$$w = \sqrt{pi/2} = \frac{pi}{2}$$​

After pulling out the 2 that comes along with the dw, this is what my integral is looking like.

$$7\frac{1}{2}\int w*cos(w) dw$$​

Now, using integration by parts...

$$u = w$$

$$du = dw$$

$$v = sinw$$

$$dv = cosw dw$$

$$uv - \int vdu$$
$$7\frac{1}{2}(w*sinw - \int sinw dw)$$
$$7\frac{1}{2}(w*sinw + cosw)$$​

Substituting back the original value of w...

$$7\frac{1}{2}(\theta^2*sin\theta^2 + cos\theta^2)$$​

Now using the Fundamental Theorem...

$$7\frac{1}{2}(\frac{pi}{2}^2*sin\frac{pi}{2}^2 + cos\frac{pi}{2}^2) - 7\frac{1}{2}(0^2*sin0^2 + cos0^2)$$​

This gives some ridiculous answer:

$$3.5 ((pi/2)^2 sin((pi/2)^2)+cos((pi/2)^2))-3.5 (0^2 sin(0^2)+cos(0^2)) = -3.5 cosh(0)+3.5 (cosh(-i (pi/2)^2)-cos(pi/2+(pi/2)^2) (pi/2)^2)$$​

...which is obviously incorrect. Where did I go wrong?

Thanks.

 

[fzero]

To make the bounds of integration to work with this substitution...

$$w = 0^2 = 0$$
$$w = \sqrt{pi/2} = \frac{pi}{2}$$​

$$7\frac{1}{2}(w*sinw + cosw)$$

Substituting back the original value of w...

Since you've already transformed the limits of integration, there's no reason to substitute back to the original variable. If you do, you need to use the original limits, not the transformed ones.

 

[tangibleLime]

Ahhh, thanks. Worked out well.

$$((7/2)((pi/2)sin(pi/2)+cos(pi/2)))-((7/2)(0*sin0+cos0))$$

= (7 pi)/4-7/2