Math Help - direct integration

1. direct integration

$\int tan(x/2).sec^2(x/2)+sec^2(x/2)$
$t=(x/2)$
$\int tan(t). sec^2(t)dt+tan(t) = -ln[cos(t)]+tan(t)+C$

But isnt correct, i'm afraid my substitution is not correct

2. Re: direct integration

$\int \tan\left(\frac{x}{2} \right)\sec^2\left(\frac{x}{2} \right)+\sec^2\left(\frac{x}{2} \right)\,dx$

I would first factor the integrand:

$\int \sec^2\left(\frac{x}{2} \right)\left(\tan\left(\frac{x}{2} \right)+1 \right)\,dx$

Now, let:

$u=\tan\left(\frac{x}{2} \right)+1\,\therefore\,du=\frac{1}{2}\sec^2\left( \frac{x}{2} \right)\,dx$

and the integral becomes:

$2\int u\,du$

Now, find the anti-derivative, then back-substitute for $u$.

3. Re: direct integration

Originally Posted by MarkFL2

$\int \tan\left(\frac{x}{2} \right)\sec^2\left(\frac{x}{2} \right)+\sec^2\left(\frac{x}{2} \right)\,dx$

I would first factor the integrand:

$\int \sec^2\left(\frac{x}{2} \right)\left(\tan\left(\frac{x}{2} \right)+1 \right)\,dx$

Now, let:

$u=\tan\left(\frac{x}{2} \right)+1\,\therefore\,du=\frac{1}{2}\sec^2\left( \frac{x}{2} \right)\,dx$

and the integral becomes:

$2\int u\,du$

Now, find the anti-derivative, then back-substitute for $u$.
MarkFL i'm learning now integral in school and we dont use substitution yet(integration techniques)
.

4. Re: direct integration

Originally Posted by Chipset3600
...i'm afraid my substitution is not correct
Originally Posted by Chipset3600
...we dont use substitution yet...
Do you see why I assumed you wanted to use a substitution?

5. Re: direct integration

Originally Posted by MarkFL2
Do you see why I assumed you wanted to use a substitution?
Yes, sorry! I did it just to ease my calculations

6. Re: direct integration

This is a difficult integral for someone who hasn't learned about u-substitution yet. You're probably aware that the derivative of $\tan{\frac{x}{2}}$ is $\frac{1}{2}\sec^2{\frac{x}{2}}$. And looking at the other term might eventually lead you to take the derivative of $\tan^2{\frac{x}{2}}$. Once you do that, you should be able to combine the two derivatives to get the expression you want to integrate.

It's probably less work to first learn u-substution, then apply it to this integral. But I guess that's not an option for you.

- Hollywood

7. Re: direct integration

Originally Posted by hollywood
This is a difficult integral for someone who hasn't learned about u-substitution yet. You're probably aware that the derivative of $\tan{\frac{x}{2}}$ is $\frac{1}{2}\sec^2{\frac{x}{2}}$. And looking at the other term might eventually lead you to take the derivative of $\tan^2{\frac{x}{2}}$. Once you do that, you should be able to combine the two derivatives to get the expression you want to integrate.

It's probably less work to first learn u-substution, then apply it to this integral. But I guess that's not an option for you.

- Hollywood
Actually isnt so hard, take a look:

$\int tan(x/2)sec^2(x/2)dx.(\frac{1}{2}.\frac{2}{1}) + \int sec^2(x/2)dx.(\frac{1}{2}.\frac{2}{1})$

$2\int tan(x/2).\frac{1}{2} sec^2(x/2)dx + 2\int sec^2(x/2). \frac{1}{2}dx$

Now i use the power rule for the first and the sec^2(u) rule for the second integral

$tan^2(x/2)+ 2tan(x/2)+C$
Now take a look of the derivative of my answer: