Tuesday, May 22, 2012

How to Use Trigonometric Substitution to Integrate

With the trigonometric substitution method, you can do integrals containing radicals of the following forms (given a is a constant and u is an expression containing x):

You’re going to love this technique … about as much as sticking a hot poker in your eye.
Before you look at how trigonometric substitution works, here are some mnemonic tricks to help you keep this method straight. Remember, with mnemonic devices, silly (and vulgar) works. First, this involves three trig functions, tangent, sine, and secant. Their initial letters, t, s, and s, are the same letters as the initial letters of the name of this technique, trigonometric substitution. Pretty nice, eh?
Okay, this sin/ass mnemonic is admittedly pretty weak. If you can come up with a better mnemonic, use it!
Now, ready to do a problem?
  1. Draw a right triangle—basically a SohCahToa triangle—where
    a SohCahToa triangle is shown in the following figure.
  2. Solve
    then differentiate, and solve for dx.
  3. Find which trig function is represented by the radical over the a
    and then solve for the radical.
    Look at the triangle in the figure. The radical is the hypotenuse and a is 2, the adjacent side, so
  4. Use the results from Steps 2 and 3 to make substitutions in the original problem and then integrate.
  5. Substitute the x expressions from Steps 1 and 3 back in for
    You can also get the expressions from the triangle in the above figure.
Was that fun or what?


Saturday, May 5, 2012

Some quotes that can help you move ahead in life

If you  need  some motivation to move ahead in your life today, try out one of these 20 encouraging quotes to get your day started.
“Twenty years from now you will be more disappointed by the things that you didn’t do than by the ones you did do. So throw off the bowlines. Sail away from the safe harbor. Catch the trade winds in your sails. Explore. Dream. Discover.”
- Mark Twain
“Get going. Move forward. Aim High. Plan a takeoff. Don’t just sit on the runway and hope someone will come along and push the airplane. It simply won’t happen. Change your attitude and gain some altitude. Believe me, you’ll love it up here.”
- Donald Trump
“I took a deep breath and listened to the old brag of my heart. I am, I am, I am.”
- Sylvia Plath
“Opportunity is missed by most people because it is dressed in overalls and looks like work.”
- Thomas A. Edison
“I decided, very early on, just to accept life unconditionally; I never expected it to do anything special for me, yet I seemed to accomplish far more than I had ever hoped. Most of the time it just happened to me without my ever seeking it.”
- Audrey Hepburn
“Knowledge has to be improved, challenged, and increased constantly, or it vanishes.”
- Peter Drucker
“If you don’t pay appropriate attention to what has your attention, it will take more of your attention than it deserves.”
- David Allen
“I’d rather be a failure at something I love than a success at something I hate.”
- George Burns
“The miracle is not that we do this work, but that we are happy to do it.”
“Learn from yesterday, live for today, hope for tomorrow. The important thing is not to stop questioning.”
- Albert Einstein
“I find hope in the darkest of days, and focus in the brightest. I do not judge the universe.”
- Dalai Lama
“Character cannot be developed in ease and quiet. Only through experience of trial and suffering can the soul be strengthened, ambition inspired, and success achieved.”
- Helen Keller
“And no, we don’t know where it will lead. We just know there’s something much bigger than any of us here.”
- Steve Jobs
“Real integrity is doing the right thing, knowing that nobody’s going to know whether you did it or not.”
- Oprah Winfrey
“Talent is cheaper than table salt. What separates the talented individual from the successful one is a lot of hard work.”
- Stephen King
“Big jobs usually go to the men who prove their ability to outgrow small ones.”
- Ralph Waldo Emerson
“If they can make penicillin out of mouldy bread, they can sure make something out of you.”
-Muhammad Ali
“Never give up, for that is just the place and time that the tide will turn.”
- Harriet Beecher Stowe
“It is by going down into the abyss that we recover the treasures of life. Where you stumble, there lies your treasure.”
- Joseph Campbell
“In essence, if we want to direct our lives, we must take control of our consistent actions. It’s not what we do once in a while that shapes our lives, but what we do consistently.”
- Tony Robbins