Special Standard Forms

Transcript

0:00Introduction

Hello everyone! One small puzzle before we start. Think about two integrals: d x over x squared plus nine, and d x over nine minus x squared. Only one sign has changed, right? But one gives tan inverse, the other a natural log. Which is which? Today: the special standard forms, plus two tricks, completing the square and splitting the top. They come up often in JEE, so let's begin.

0:30Squares in the denominator

You already know the integral of d x over one plus x squared: tan inverse x. Now put a squared in place of one, where a is a positive constant. The long S is the integral sign, and d x says the variable is x. We get one over a, times tan inverse of x over a, plus capital C, the constant of integration. The little minus one on tan is not a power; it means inverse, okay? Now change the plus to a minus: d x over x squared minus a squared. Does tan inverse still work here? No, it does not! With a minus sign, we get a log. One over two a, times natural log of mod of x minus a, over x plus a, plus capital C, as always. The l n is the natural log, and the two straight bars are the modulus. Right? And the third: d x over a squared minus x squared. Careful, the order inside the log flips: one over two a, times natural log of modulus of a plus x, over a minus x, plus capital C, the constant. Copy the previous formula here, and your answer has the wrong sign. Got it?

1:40Square roots: sine inverse and a log

Now a square root in the denominator; the tick-shaped sign is the square root. You know that d x over root of one minus x squared gives sine inverse x. So what does d x over root of a squared minus x squared give? Sine inverse of x over a, plus capital C, correct! But look: no one over a in front this time. Students often add it out of habit, so watch out. Next, d x over root of x squared plus or minus a squared. The plus-minus sign covers both cases at once. We get natural log of modulus of x, plus root of x squared plus or minus a squared, plus capital C, the constant. The sign under the root stays the same in the answer. Clear? Now the root moves to the top. The integral of root of a squared minus x squared, d x, is x over two times that root, plus a squared over two times sine inverse of x over a, plus capital C, okay? And for root of x squared plus or minus a squared: x over two times that root, then plus or minus a squared over two, times natural log of modulus of x plus that root, plus capital C, as always. The sign in the middle follows the sign under the root. Learn these by heart.

3:01A quadratic in the denominator

Often the denominator is a full quadratic, a x squared plus b x plus c, with or without a root. Here a, b and c are just its coefficients, not the a of our formulas, okay? So how do we turn this into one of our standard forms? Yes, exactly! Complete the square. For example, x squared plus two x plus five equals x plus one, whole squared, plus four. Half of two is one, and five minus one leaves four. Right? Four is two squared, so this is the tan inverse form, with x plus one in place of x, and a equals two. Since d of x plus one is d x, it works directly. Got it?

3:45Linear over quadratic: split the top

Now the last family: p x plus q on top, over the same quadratic, or over its square root. Here p and q are numbers, okay? First, tell me: what is the derivative of a x squared plus b x plus c? Two a x plus b. Very good! This is what we want on top. So write p x plus q as lambda times two a x plus b, plus mu. Lambda and mu, two Greek letters, are constants; compare coefficients to find them. With me? Then split the integral. In the lambda part, the top is the derivative of the bottom, so put t equals the quadratic. In the mu part, complete the square, as before. Clear?

4:31Example 1: 1 over x2 + 4x + 13, and 1 over 9 - x2

Example one, from the course. Find the integral of d x over x squared plus four x plus thirteen, and of d x over nine minus x squared. Pause the video and try it first. Done? Okay, let's check. Complete the square: half of four is two, and x plus two, whole squared, gives x squared plus four x plus four. Thirteen minus four leaves nine, which is three squared. So which standard form is this? Yes, exactly! Plus sign, so tan inverse, with x plus two in place of x, and a equals three. One third, times tan inverse of x plus two over three, plus capital C, okay? Now nine minus x squared, the minus partner from our puzzle! Nine is three squared, so it is a squared minus x squared, with a equals three. Minus sign, so a log, and two a is six. One sixth, times natural log of modulus of three plus x, over three minus x, plus capital C, as always. Writing x minus three over x plus three is the trap; that order is for x squared minus a squared. So our puzzle is solved, right?

5:46Example 2: x + 2 over root of x2 + 2x + 3, split the top

Now a harder one, Example two. Find the integral of x plus two, over root of x squared plus two x plus three, d x. Pause the video and try it first. Done? Okay, let's check. The derivative of x squared plus two x plus three is two x plus two. We want that on the top. So write x plus two as one half times two x plus two, plus one. Check: half of two x plus two is x plus one, plus one is x plus two. Perfect! So lambda is one half, and mu is one. Now the lambda part. The top is the derivative of the bottom, so the answer is a log, right? No, no, careful here! The bottom has a square root. Put t equals x squared plus two x plus three, so d t equals two x plus two, d x. One half times the integral of d t over root t is one half times two root t, which is root t, the root of the quadratic. Clear? Now the mu part, d x over the same root. Complete the square: x squared plus two x plus three equals x plus one, whole squared, plus two. So a is root two. Okay? The plus form under a root gives a log: natural log of modulus of x plus one, plus the root. That root is root of x squared plus two x plus three again. Right? Add the two parts; one capital C at the end covers both. Root of x squared plus two x plus three, plus natural log of modulus of x plus one plus that root, plus capital C, the constant. So, split the top, and each piece is a standard form.

7:31Quick revision: tan inverse, log, sin inverse, split top

So, quick revision. With squares, plus gives tan inverse and minus gives a log. Under a root, it is sine inverse or a log. Full quadratic at the bottom? Complete the square first. Linear on top? Write it as lambda times the derivative, plus mu, and split. And always add plus capital C, okay? Now go and try the practice problems on this page. All the best!