How can I integrate trig functions using u-substitution and n-substitution?

[marc017]

This isn't homework, It is just book problems that I am practicing, I am checking some answers with wolfram and others with the book answers.

Homework Statement

$$\begin{align} \int \frac{sin(w)\,dw}{\sqrt{1-cos(w)}}\\ \end{align}$$

Homework Equations

I used u substitution... Not sure if I approached this problem the correct way

The Attempt at a Solution

$$\begin{align} \int \frac{sin(w)\,dw}{\sqrt{1-cos(w)}}\\ \end{align}$$

Using U sub... U = cos(w), du = -sin(w)

$$\begin{align} - \int \frac{\,du}{\sqrt{1-u}}\\ \end{align}$$

Using n sub... n=1-u, dn = -1

$$\begin{align} \int \frac{\,du}{\sqrt{n}} = 2\sqrt{(1-cos(w))} + C\\ \end{align}$$

 

[Mark44]

This isn't homework, It is just book problems that I am practicing, I am checking some answers with wolfram and others with the book answers.

Homework Statement

$$\begin{align} \int \frac{sin(w)\,dw}{\sqrt{1-cos(w)}}\\ \end{align}$$

Homework Equations

I used u substitution... Not sure if I approached this problem the correct way

The Attempt at a Solution

$$\begin{align} \int \frac{sin(w)\,dw}{\sqrt{1-cos(w)}}\\ \end{align}$$

Using U sub... U = cos(w), du = -sin(w)

$$\begin{align} - \int \frac{\,du}{\sqrt{1-u}}\\ \end{align}$$

Using n sub... n=1-u, dn = -1

$$\begin{align} \int \frac{\,du}{\sqrt{n}} = 2\sqrt{(1-cos(w))} + C\\ \end{align}$$

Looks good. One nice thing about these types of problems is that you can check them yourself. If your answer is correct, you should be able to differentiate it and get the integrand.

As for your substitution, what you did is OK, but you can kill two birds with one stone by letting u = 1 - cos(w). Then du = sin(w)dw.

 

[marc017]

Looks good. One nice thing about these types of problems is that you can check them yourself. If your answer is correct, you should be able to differentiate it and get the integrand.

As for your substitution, what you did is OK, but you can kill two birds with one stone by letting u = 1 - cos(w). Then du = sin(w)dw.

Thank you.. I was checking the integrals on wolfram but it seems to work much better if I take my answer and try to get the integral. And I can't believe I didn't think of the 1-cos(w) substitution

I just found this forum and you guys have been a lot of help! Maybe one day i will be good enough to answer other people's questions.

 

[SammyS]

Thank you.. I was checking the integrals on wolfram but it seems to work much better if I take my answer and try to get the integral.
...

Speaking of using WolframAlpha, their result for this integration is $\displaystyle \ \frac{4 \sin^2(x/2)}{(1-\cos(x))^{3/2}}+C \ .$

By the way, this is equivalent to your answer.