1. Xtra Edge February 2012 - Career Point

dx
(ii) ∫ 2
ax + bx + c
This can be reduced to one of the forms of the above
formulae (15), (20) or (21).
(iii) ∫ + bx + c dx
ax 2
This can be reduced to one of the forms of the above
formulae (22), (23) or (24).
( px + q)
dx ( px + q)
dx
(iv) ∫ ,
2
ax + bx + c ∫ 2
ax + bx + c
For the evaluation of any of these integrals, put
px + q = A {differentiation of (ax 2 + bx + c)} + B
Find A and B by comparing the coefficients of like
powers of x on the two sides.
1. If k is a constant, then
∫ dx
k = kx and ∫ k f ( x)
dx = k∫ f ( x)
dx
2. ∫ f ( x)
± f ( x)}
dx
{ 1 2 = ∫ f 1 ( x)
dx ± ∫ f 2 ( x)
dx
Some Proper Substitutions :
1. ∫ f(ax + b) dx, ax + b = t
2. ∫ f(axn + b)x n–1 dx, ax n + b = t
3. ∫ f{φ(x)} φ´(x) dx, φ(x) = t
4.
f ´( x)
∫ dx , f(x) = t
f ( x)
5. ∫ − 2 2
x
6. ∫ + 2 2
x
a dx, x = a sin θ or a cos θ
a dx, x = a tan θ
2
2
7. ∫
a − x
2 2
a + x
dx, x 2 = a 2 cos 2θ
8. ∫ a ± x dx, a ± x = t 2
a − x
9. ∫ dx, x = a cos 2θ
a + x
10. ∫ − 2
2ax x dx, x = a(1 – cos θ)
11. ∫ − 2 2
a
x dx, x = a sec θ
Substitution for Some irrational Functions :
dx
1. ∫ , ax + b = t
( px + q)
ax + b
2
dx
2. ∫ 2
( px + q)
ax + bx + c
1
, px + q =
t
dx
3. ∫ , ax + b = t
2
( px + qx + r)
ax + b
2
dx
1 2 2
4. ∫ , at first x = and then a + ct = z
2
2
( px + r)
ax + c
t
Some Important Integrals :
dx
⎛ x − α ⎞
1. To evaluate ∫ ,
( x − α)(
x − β)
∫ ⎜ ⎟ dx,
⎝ β − x ⎠
XtraEdge for IIT-JEE 43 FEBRUARY 2012
∫
( x − α)(
β − x)
dx. Put x = α cos 2 θ + β sin 2 θ
2.
dx dx
To evaluate ∫ ,
a + b cos x ∫ a + bsin
x
,
dx
∫ a + b cos x + c sin x
⎛ x ⎞
⎛ 2 x ⎞
⎜2
tan ⎟
⎜1−
tan ⎟
Replace sin x =
⎝ 2 ⎠
and cos x =
⎝ 2 ⎠
⎛ 2 x ⎞
⎛ 2 x ⎞
⎜1+
tan ⎟
⎜1+
tan ⎟
⎝ 2 ⎠
⎝ 2 ⎠
x
Then put tan = t.
2
3.
p cos x + q sin x
To evaluate ∫ dx
a + b cos x + c sin x
Put p cos x + q sin x = A(a + b cos x + c sin x)
+ B. diff. of (a + b cos x + c sin x) + C
A, B and C can be calculated by equating the
coefficients of cos x, sin x and the constant terms.
4. To evaluate
dx
∫ ,
2
2
a cos x + 2b
sin x cos x + c sin x
dx
dx
∫ ,
2
a cos x + b ∫ 2
a + bsin
x
In the above type of questions divide N r and D r by
cos 2 x. The numerator will become sec 2 x and in the
denominator we will have a quadratic equation in tan
x (change sec 2 x into 1 + tan 2 x).
Putting tan x = t the question will reduce to the form
dt
∫ 2
at + bt + c
5. Integration of rational function of the given form
x + a
(i) ∫ 4 2
x + kx + a
2
2
4
x − a
dx, (ii) ∫ 4 2
x + kx + a
2
2
4
dx, where
k is a constant, positive, negative or zero.
These integrals can be obtained by dividing
numerator and denominator by x 2 , then putting
a
x –
x
2
a
= t and x +
x
2
= t respectively.
Integration of Product of Two Functions :
1. ∫ f1(x) f2(x) dx = f1(x) ∫ f2(x) dx –
' [ ( f 1 ( x)
f2
( x)
] dx
∫ ∫ dx
Proper choice of the first and second functions :
Integration with the help of the above rule is called