ATAR Notes: Forum

VCE Stuff => VCE Mathematics => VCE Mathematics/Science/Technology => VCE Subjects + Help => VCE Specialist Mathematics => Topic started by: Electroflame1395 on August 13, 2013, 07:37:25 pm

Title: Integral Calculus HELP!!!
Post by: Electroflame1395 on August 13, 2013, 07:37:25 pm: Hey, i have got a few question that i need help with. It would be awesome if anyone can help me.
I have save it onto a word document.
For 2) i am pretty sure you have to separate it into tan(x)*tan^2(x) but i could not get any further.
I had could not any where with the rest.
THANKS
Title: Re: Integral Calculus HELP!!!
Post by: SocialRhubarb on August 13, 2013, 08:01:20 pm: Question 2:
$\int{\tan^3(x)}dx$

$=\int{\frac{\sin^3(x)}{\cos^3(x)}}dx$

$\text{Let u = cos(x)}, \frac{du}{dx}=-\sin(x)$

$\int{\frac{\sin^3(x)}{\cos^3(x)}}dx$

$=-\int{\frac{\sin^2(x)\frac{du}{dx}}{u^3}}dx$

$=-\int{\frac{1-\cos^2(x)}{u^3}}du$

$=-\int{(\frac{1}{u^3}-\frac{u^2}{u^3})}du$

$=\int{(-\frac{1}{u^3}+\frac{1}{u})}du$

$=\frac{1}{2u^2}+\ln(u)+C$

$=\frac{1}{2}\sec^2(x)+\ln|\cos(x)|+C$

There may be a simpler solution using the identity $\tan^2(x)+1=\sec^2(x)$ . Did you just want the solutions to question 2?

EDIT: left out some negatives.
Title: Re: Integral Calculus HELP!!!
Post by: abeybaby on August 13, 2013, 08:18:23 pm: Here you go, good luck!

EDIT: Question 2, left out the modulus signs for the log. sorry!
Title: Re: Integral Calculus HELP!!!
Post by: Electroflame1395 on August 13, 2013, 10:08:26 pm: Quote from: SocialRhubarb on August 13, 2013, 08:01:20 pm
Question 2:
$\int{\tan^3(x)}dx$

$=\int{\frac{\sin^3(x)}{\cos^3(x)}}dx$

$\text{Let u = cos(x)}, \frac{du}{dx}=-\sin(x)$

$\int{\frac{\sin^3(x)}{\cos^3(x)}}dx$

$=-\int{\frac{\sin^2(x)\frac{du}{dx}}{u^3}}dx$

$=-\int{\frac{1-\cos^2(x)}{u^3}}du$

$=-\int{(\frac{1}{u^3}-\frac{u^2}{u^3})}du$

$=\int{(-\frac{1}{u^3}+\frac{1}{u})}du$

$=\frac{1}{2u^2}+\ln(u)+C$

$=\frac{1}{2}\sec^2(x)+\ln|\cos(x)|+C$

There may be a simpler solution using the identity $\tan^2(x)+1=\sec^2(x)$ . Did you just want the solutions to question 2?

EDIT: left out some negatives.

Thanks a lot but actually i needed help with the whole lot. Please if you are able to solve the rest it would be great.
Title: Re: Integral Calculus HELP!!!
Post by: Electroflame1395 on August 13, 2013, 10:12:33 pm: Quote from: abeybaby on August 13, 2013, 08:18:23 pm
Here you go, good luck!

EDIT: Question 2, left out the modulus signs for the log. sorry!

Haha thanks, you have made my night. I did not know that you had attached a file and answered all the questions.
Thanks you so much.
Title: Re: Integral Calculus HELP!!!
Post by: abeybaby on August 13, 2013, 11:13:17 pm: Quote from: Electroflame1395 on August 13, 2013, 10:12:33 pm
Haha thanks, you have made my night. I did not know that you had attached a file and answered all the questions.
Thanks you so much.
no problem!