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June 3rd, 2017, 09:43 AM  #1 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  Random calculus practice problems
Hi guys, Is there a software or website where I can practice random problems? I don't like using the book problems because when I do them I know exactly in which section I am and that there's only one way to do it. For example, all the problems related to integration by parts would be in the section for integration part. So I already know that. I want something that will allow me to solve the problem without already knowing the section. Let me know please. Last edited by skipjack; June 3rd, 2017 at 11:02 AM. 
June 3rd, 2017, 04:49 PM  #2 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8 
anyone? I know you guys practice a lot 
June 3rd, 2017, 05:34 PM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,614 Thanks: 2603 Math Focus: Mainly analysis and algebra  
June 3rd, 2017, 07:05 PM  #4 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1122 Math Focus: Elementary mathematics and beyond 
Not an easy problem, perhaps. $\displaystyle \int_{0}^{\pi} \dfrac{2\sin x+3\cos x3}{13\cos x5} \,dx$ 
June 3rd, 2017, 09:55 PM  #5 
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  I just can't see it sir. Last edited by skipjack; June 6th, 2017 at 10:58 PM. 
June 6th, 2017, 08:08 PM  #6  
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1122 Math Focus: Elementary mathematics and beyond  Quote:
Then $$\cos^2\frac x2=\frac{1+\cos x}{2}\Rightarrow\frac{1}{1+t^2}=\frac{1+\cos x}{2}\implies\cos x=\frac{1t^2}{1+t^2}$$ $$\sin x=2\sin\frac x2\cos\frac x2\Rightarrow\sin^2x=4\sin^2\frac x2\cos^2\frac x2=4\frac{1\cos^2\frac x2}{1+t^2}=\frac{4t^2}{(1+t^2)^2}\implies\sin x=\frac{2t}{1+t^2}$$ and $$dt=\frac{\sec^2\frac x2}{2}\,dx\implies\frac{2dt}{1+t^2}=dx$$ Now, using the above, make the necessary substitutions to express the integral in terms of $t$, simplify, apply partial fractions and adjust the bounds of integration. The resulting improper integral may be integrated directly, then the bounds may be applied. For more information about the substitution used, see here. Last edited by greg1313; June 6th, 2017 at 08:12 PM.  
June 7th, 2017, 01:33 PM  #7  
Member Joined: Jan 2017 From: California Posts: 80 Thanks: 8  Quote:
 
June 7th, 2017, 07:05 PM  #8  
Math Team Joined: Dec 2013 From: Colombia Posts: 7,614 Thanks: 2603 Math Focus: Mainly analysis and algebra  Quote:
\begin{align*} \sin x &= 2\sin \tfrac x2 \cos \tfrac x2 &&= 2\frac{\sin \tfrac x2}{\cos \tfrac x2} \cos^2 \tfrac x2 \\ &= \frac{2\tan \tfrac x2}{\sec^2 \tfrac x2} &&= \frac{2\tan \tfrac x2}{1 + \tan^2 \tfrac x2} \\ &= \frac{2t}{1 + t^2} \end{align*}  
June 8th, 2017, 09:30 AM  #9 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,923 Thanks: 1122 Math Focus: Elementary mathematics and beyond  

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