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 .
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.
If g is a differentiable function of x, then
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.
In the integration of a function, if the integrand involves any kind of trigonometric function, then we use trigonometric identities to simplify the function that can be easily integrated.
Few of the trigonometric identities are as follows:
All these identities simplify integrand, that can be easily found out.
Ex.1 Evaluate (x^{2} +1)^{2} (2x) dx .
Sol. Letting g(x) = x^{2} + 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.2 Evaluate
Sol.
Ex.3 Evaluate
Sol.
Let u = x^{4 }+ 2 ⇒ du = 4x^{3} dx
Ex.4 Evaluate
Sol.
Let u = x^{3} – 2. Then du = 3x^{2} dx. so by substitution :
Ex.5 Evaluate
Sol. Let u = . Then u^{2 }= x + 4, so x = u^{2 }–4 and dx = 2u du.
Therefore
Ex.6 Evaluate
Sol. Rewrite the integrand as follows :
= – ln (e^{x} + 1) + c (∴ e^{x} + 1 > 0)
Ex.7 Evaluate sec x dx
Sol. Multiply the integrand sec x by sec x + tan x and divide by the same quantity :
Ex.8 Evaluate cos x (4  sin2 x) dx
Sol. Put sin x = t so that cos x dx = dt. Then the given integral =
Ex.9 Integrate
(i)
(ii)
Sol.
Ex.10 Integrate
(i)
(ii)
Sol.
Ex.11 Integrate
(i)
(ii)
Sol.
Ex.12 Integrate
Sol.
Ex.13 Integrate cos^{5}x.
Sol.
[put sin x = t ⇒ cos x dx = dt]
Ex.14 Evaluate
Sol.
Ex.15 Integrate 1/(sin^{3} x cos^{5}x).
Sol. Here the integrand is sin^{–3} x cos^{–5}x. It is of type sin^{m} x cos^{n} x,where m + n = –3 –5 = –8 i.e., –ve even integer
Now put tan x = t so that sec^{2 }x dx = dt
Ex.16 Integrate
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.
,putting tan x = t and sec^{2}x dx = dt
Ex.17 Evaluate
Sol.
Put x – β = y ⇒ dx = dy
Given integral
Now put sinθ + cosθ tan y = z^{2} ⇒ cosq sec^{2} y dy = 2z dz
Ex.18 Evaluate dx
Sol.
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1. What is integration by substitution? 
2. How do you perform integration by substitution? 
3. What are trigonometric identities used for in integration? 
4. How can trigonometric identities be applied in integration? 
5. Can integration by substitution and trigonometric identities be used together? 

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