• BLOG
• ABOUT
• LECTURER
• OLYMPIAD
• S-PAPER
• PURE APPLIED
• MISC.
• CONTACT

I'm not sure about you guys, but I'm a huge Prison Break Fan. For those who are unfamiliar with the series, the story is about a civil engineer named Michael Scofield who impersonates himself in Fox River penitentiary so that he can break out with his brother Lincoln Borrows, who was framed for a crime he did not commit. Throughout their escape, they are bombarded with misfortunates and setbacks that would make any escapee give up. Despite all this, they still progress in their escape on the basis of what Michael says time and time again ‘All you need is a little bit of faith.’

When solving various difficult questions in mathematics, it is that little bit of faith that brings you to the answer. Agreed that at times the method of approach in answering the question may be wrong. However, we never know whether we are using the right method unless we have gone through it. Sadly many of us don’t because the barriers encountered in a particular method turn us down. It is here where faith brings carries us through. For me, it was a trigonometric integration question that taught me the importance of faith.

I was given the daunting task to evaluate  which I would say isn’t your standard calculus problem. Well, I could easily see that  hidden in the integrand so using trigonometric identities, I had

At this point, I’m already questioned myself whether I made the right move. I had seemed to have a more complicated expression. Well, I evaluated the left integral first to see what I got.

Recognizing that I could use the standard integral of  to help me evaluate , I multiplied top and bottom by a certain expression to get

 

To my surprise, the derivative of the denominator nicely equaled the numerator. So, I ultimately had

First half done. Now, all I have to deal with is the . It initially looked fairly easy as I could create a standard form with careful manipulation of the derivatives of  and . However, no matter how hard I tried, I couldn't express the integrand in the form . I was about to give up. It looked like my intuitive method in the previous step was a waste as it seem impossible to evaluate .

Just have a little bit of faith, I told myself. This could be the right method in solving the question. I didn’t know. But I knew that I needed faith to believe I would arrive to the right answer.

I tried integration by parts to see what I have. Hmmm, I could write the integral as

and then applying the by parts formula to get

Another waste of effort. I ended up with  again and probably lost all chances of evaluating the integral. Faith was a failure as I ventured into deeper hole. Or was it a hole with light at the end?

Wait a minute. I could bring the  to the other side of the equation, combine it with the  over there and basically have the solution on the right hand side like such.

Faith brought me to the light. See, throughout solving the problem, I did in fact use the right method, thought it was filled with difficulties and detours which discouraged me from proceeding. There were a few turns at which I wanted to give up. So I needed faith. I needed faith to tell me that this was the correct method and despite the difficulties, it was in my best interest in persevere and handle each calculation as it came by because there was that lingering belief that this method could be correct.

All I need was that little bit of faith, and without it, I would still be stuck in finding another method which would be futile.