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waterdemon

Here is what you got to do when integration has been asked of a particular type of expression :

**f(ax+b)

put (ax+b)=t

**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.

**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

**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.

** 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.

**1/aSinx+bCosx

We take a=rCos@ and b=rSin@

as r=(a²+b²)^1/2.@=tan^-1(b/a)

Now solve.

**aSinx+bCosx/cSinx+dCosx

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

Get value of l and m and solve.

**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 + m

Diff. of Den/Den. + n

1/pSinx+qCosx+r

Now solve.

**

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

**

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

**

(px+q)(ax²+bx+c)^1/2.dx

px+q=l(Diff. of ax²+bx+c)+m

Find l and m and solve.

**

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

*P and Q are linear.

put Q =t².

Gor example of a form.

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

put (cx+d)=t².

*P is quadratic and Q is linear

1/(ax²+bx+c)(dx+e)^1/2.dx

put (dx+e)=t².

*P is linear and Q is Quadratic.

1/(ax+b)(cx²+dx+e)^1/2.dx

put (ax+b)=1/t

*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.

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