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# Integration by Substitution JEE Notes | EduRev

## JEE : Integration by Substitution JEE Notes | EduRev

The document Integration by Substitution JEE Notes | EduRev is a part of the JEE Course Mathematics (Maths) Class 12.
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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  f(g(x))g'(x) dx = F(g(x)) + C.
If u = g(x), then du = g'(x) and  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

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  then the substitution u =  may be effective.

SOME STANDARD SUBSTITUTIONS

Ex.8 Evaluate  (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

Ex.9 Evaluate
Sol.

Ex.10 Evaluate
Sol.

Let u = x+ 2   â‡’   du = 4x3 dx

Ex.11 Evaluate

Sol.

Let u = x3 â€“ 2. Then du = 3x2 dx. so by substitution :

Ex.12 Evaluate

Sol. Let u  =   . Then u= x + 4, so x = uâ€“4 and dx = 2u du.

Therefore

Ex.13 Evaluate

Sol. Rewrite the integrand as follows :

= â€“ ln (e-x + 1) + c   (âˆ´  e-x + 1 > 0)

Ex.14 Evaluate  sec x dx

Sol. Multiply the integrand sec x by sec x + tan x and divide by the same quantity :

Ex.15 Evaluate cos x (4 - sin2 x) dx
Sol. Put sin x = t so that cos x dx = dt. Then the given integral =

Ex.16 Integrate

(i)
(ii)

Sol.

Ex.17 Integrate

(i)
(ii)

Sol.

Ex.18 Integrate

(i)
(ii)

Sol.

Ex.19 Integrate

Sol.

Ex.20 Integrate cos5x.

Sol.

[put sin x = t â‡’ cos x dx = dt]

Ex.21 Evaluate

Sol.

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

Sol. Here the integrand is sinâ€“3 x cosâ€“5x. It is of type sinm x cosn x,where m + n = â€“3 â€“5 = â€“8 i.e., â€“ve even integer

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

Ex.23 Integrate

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

,putting tan x = t and sec2x dx = dt

Ex.24 Evaluate

Sol.

Put x â€“ Î² = y  â‡’  dx = dy

Given integral

Now put sinÎ¸ + cosÎ¸ tan y = z2 â‡’  cosq sec2 y dy = 2z dz

Ex.25 Evaluate dx

Sol.

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## Mathematics (Maths) Class 12

209 videos|222 docs|124 tests

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