IIT JEE , AIEEE Forum - Mathematics , Integral Calculus

roysasi

what are the shortcuts in integrals?

waterdemon

I hope these prove useful for you.

1)f(ax+b)

put (ax+b)=t

2)px+q/ax²+bx+c

take px+q=L(differentiation of ax²+bx+c) + m

Find values of L and m and substitute them in the exp.

and divide the whole by (ax²+bx+c) and now you can do

its integration easily as it is divided in two parts.

3)P(x)/ax²+bx+c

In such cases we divide the numerator by denominator

and then express as

Q(x) + [R(x)/ax²+bx+c]

Here Q(x) = Quotient after dividing.

and R(x) = Remainder after dividing.

This becomes Q(x).dx + R(x)/ax²+bx+c .dx

4)px+q/(ax²+bx+c)^1/2

Take px+q = l(Diff. of Denominator) + m

Find values of "L" and "m" and solve.

After this take x common and make the coefficient of x²

as unity.

Now add and substract the square of half of the

coefficient of x.

Now apply formula to get the answer.

5) 1/aSinx+bCosx , 1/a+bSinx , 1/a+bCosx.dx

We proceed as follows.

Take Sinx = 2 Tan x/2 / 1+tan²x/2 or

Take Cosx = 1-tan²x/2 / 1+tan²x/2

Replace 1+tan²x/2 in numerator by sec²x/2.

put tan x/2 = t as 1/2Sec²x/2.dx=dt

And solve.

6)1/aSinx+bCosx

We take a=rCos@ and b=rSin@

as r=(a²+b²)^1/2.@=tan^-1(b/a)

Now solve.

7)aSinx+bCosx/cSinx+dCosx

Numerator=l(Diff. of Denominator) + m(denominator)

Get value of L and m and solve.

8)aSinx+bCosx+c/pSinx+qCosx+r

Here c and r = Integers(constants)

Num=L(den.)+m(Diff. of Den.)+ n

Find values of L,m,n and then take the form as:

aSinx+bCosx+c/pSinx+qCosx+r

=L.dx + mDiff. of Den/Den. + n1/pSinx+qCosx+r

Now solve.

9)f'(x)/f(x).dx = log{f(x)}

10)e^x{f(x)+f'(x)}.dx = e^x{f(x)} + c

11)(px+q)(ax²+bx+c)^1/2.dx

px+q=L(Diff. of ax²+bx+c)+m

Find L and m and solve.

12)h(x)/P(Q)^1/2.dx

P and Q are linear.

put Q =t².

Got example of a form.

a)1/(ax+b)(cx+d)^1/2.dx

put (cx+d)=t².

b)P is quadratic and Q is linear

1/(ax²+bx+c)(dx+e)^1/2.dx

put (dx+e)=t².

c)P is linear and Q is Quadratic.

1/(ax+b)(cx²+dx+e)^1/2.dx

put (ax+b)=1/t

d)P and Q both are Quadratic.

1/(ax²+b)(cx²+d)^1/2.dx

put x=1/t and then c+dt² = u².

Hope you find it useful.

Rate me if useful.

Cheers!!!!!!!!!!!

astrojith

Dude, nice resource. But I didn't quite understand 5th,6th and 12(D)th points.

I think it'll be much more understood if you can either use some formatting or explain it further with an example problem.

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