Integration by Substitution JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Integration by Substitution JEE Notes | EduRev

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C. INTEGRATION BY SUBSTITUTION
Let g be a function whose range is an interval l, and let f be a function that is continuous on l. If g is differentiable on its domain and F is an antiderivative of f on l, then Integration by Substitution JEE Notes | EduRev f(g(x))g'(x) dx = F(g(x)) + C.
If u = g(x), then du = g'(x) and Integration by Substitution JEE Notes | EduRev f(u) du = F(u) + C .


GUIDELINES FOR MAKING A CHANGE OF VARIABLE
1. Choose a substitution u = g(x). Usually, it is best to choose the inner part of a composite function, such as a quantity raised to a power.
2. Compute du = g'(x) dx.
3. Rewrite the integral in terms of the variable u.
4. Evaluate the resulting integral in terms of u.
5. Replace u by g(x) to obtain an antiderivative in terms of x.

THE GENERAL POSER RULE FOR INTEGRATION
If g is a differentiable function of x, then  Integration by Substitution JEE Notes | EduRev


RATIONALIZING SUBSTITUTIONS
Some irrational functions can be changed into rational functions by means of appropriate substitutions.
In particular, when an integrand contains an expression of the form Integration by Substitution JEE Notes | EduRev then the substitution u = Integration by Substitution JEE Notes | EduRev may be effective.


SOME STANDARD SUBSTITUTIONS
Integration by Substitution JEE Notes | EduRev

Ex.8 Evaluate Integration by Substitution JEE Notes | EduRev (x2 +1)2 (2x) dx .
Sol. Letting g(x) = x2 + 1, we obtain g'(x) = 2x and f(g(x)) = [g(x)]2.

From this, we can recognize that the integrand and follows the f(g(x)) g'(x) pattern. Thus, we can write Integration by Substitution JEE Notes | EduRev

Ex.9 Evaluate  Integration by Substitution JEE Notes | EduRev
Sol.  Integration by Substitution JEE Notes | EduRev

Ex.10 Evaluate Integration by Substitution JEE Notes | EduRev
Sol.

Let u = x+ 2   ⇒   du = 4x3 dx

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Ex.11 Evaluate Integration by Substitution JEE Notes | EduRev

Sol.

Let u = x3 – 2. Then du = 3x2 dx. so by substitution :

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev
Ex.12 Evaluate  Integration by Substitution JEE Notes | EduRev

Sol. Let u  =  Integration by Substitution JEE Notes | EduRev . Then u= x + 4, so x = u–4 and dx = 2u du.

Therefore  Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev


Ex.13 Evaluate  Integration by Substitution JEE Notes | EduRev

Sol. Rewrite the integrand as follows :

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev = – ln (e-x + 1) + c   (∴  e-x + 1 > 0)


Ex.14 Evaluate  Integration by Substitution JEE Notes | EduRevsec x dx

Sol. Multiply the integrand sec x by sec x + tan x and divide by the same quantity :
Integration by Substitution JEE Notes | EduRev

 

Ex.15 Evaluate Integration by Substitution JEE Notes | EduRevcos x (4 - sin2 x) dx
Sol. Put sin x = t so that cos x dx = dt. Then the given integral = Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev


Ex.16 Integrate 

(i) Integration by Substitution JEE Notes | EduRev
(ii) Integration by Substitution JEE Notes | EduRev


Sol.

Integration by Substitution JEE Notes | EduRev


Ex.17 Integrate 

(i) Integration by Substitution JEE Notes | EduRev
(ii) Integration by Substitution JEE Notes | EduRev


Sol.

Integration by Substitution JEE Notes | EduRev


Ex.18 Integrate 

(i) Integration by Substitution JEE Notes | EduRev
(ii) Integration by Substitution JEE Notes | EduRev


Sol.

Integration by Substitution JEE Notes | EduRev

 

Ex.19 Integrate Integration by Substitution JEE Notes | EduRev


Sol.

Integration by Substitution JEE Notes | EduRev

Ex.20 Integrate cos5x.

Sol.

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev [put sin x = t ⇒ cos x dx = dt] 

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Ex.21 Evaluate Integration by Substitution JEE Notes | EduRev

Sol.

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Ex.22 Integrate 1/(sin3 x cos5x).

Sol. Here the integrand is sin–3 x cos–5x. It is of type sinm x cosn x,where m + n = –3 –5 = –8 i.e., –ve even integer

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Now put tan x = t so that secx dx = dt 

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Ex.23 Integrate Integration by Substitution JEE Notes | EduRev


Sol. Here the integrand is of the type cosm x sinnx. We have m = –3/2, n = – 5/2, m + n = – 4 i.e., and even negative integer. 

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev ,putting tan x = t and sec2x dx = dt

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Ex.24 Evaluate  Integration by Substitution JEE Notes | EduRev

Sol.

Put x – β = y  ⇒  dx = dy

Given integral 

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Now put sinθ + cosθ tan y = z2 ⇒  cosq sec2 y dy = 2z dz

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

 

Ex.25 Evaluate Integration by Substitution JEE Notes | EduRevdx

Sol.

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

Integration by Substitution JEE Notes | EduRev

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