notations of an integral

[Vali]

The last question for today.
\(\displaystyle \int_{0}^{\frac{\pi }{4}}\frac{1}{cos^{4}(x)}+\frac{1}{cos^{2}(x)}=\int_{0}^{\frac{\pi }{4}}\frac{tan^{2}(x)}{cos^{2}(x)}\)
If I note u=tan^2(x) => tan(x)=sqrt(u) I get \(\displaystyle \int_{0}^{\frac{\pi }{4}}\frac{u}{cos^{2}(x)}\cdot \frac{cos^{2}(x)}{2\sqrt{u}}du=\frac{1}{2}\cdot \frac{u^{\frac{1}{2}}}{\frac{1}{2}}=\frac{1}{2}\cdot 2\cdot 1=1\)
On the other hand, if I note u=tan(x) I get u^3/3 and when I replace with pi/4 and 0 I get 1/3 ( the right answer)
Why the results are different ? What's the difference between these notations?

 

[Dr.Peterson]

It's not a matter of notation, just of carelessness. The first is wrong because you integrated wrong (or maybe just didn't write enough steps to see what you were doing).

You have [MATH]\int\frac{u}{2\sqrt{u}}du = \int\frac{1}{2}u^{\frac{1}{2}}du[/MATH]. Continue.

You also failed to convert the limits of integration to u, but you ended up doing what would have been correct.

 

[mmm4444bot]

\(\displaystyle \frac{1}{cos^{4}(x)}+\frac{1}{cos^{2}(x)}\)

Is the addition supposed to be a subtraction?

?

 

[Vali]

Yes, my bad.

 

[pka]

\(\displaystyle \int_{0}^{\frac{\pi }{4}}\frac{1}{cos^{4}(x)}+\frac{1}{cos^{2}(x)}=\int_{0}^{\frac{\pi }{4}}\frac{tan^{2}(x)}{cos^{2}(x)}\)

Is the addition supposed to be a subtraction?

Yes, my bad.

Take note: \(\displaystyle \frac{1}{\cos^4(x)}-\frac{1}{\cos^{2}(x)}=\frac{\sec^2(x)}{\cos^{2}(x)}-\frac{1}{\cos^{2}(x)}=\frac{\tan^2(x)}{\cos^2(x)}\)