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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 **** (x ^{2} +1)^{2} (2x) dx .**

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^{4 }+ 2 â‡’ du = 4x^{3} dx

**Ex.11 Evaluate **

**Sol.**

Let u = x^{3} â€“ 2. Then du = 3x^{2} dx. so by substitution :

**Ex.12 Evaluate **

**Sol.** Let u = . Then u^{2 }= x + 4, so x = u^{2 }â€“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 cos ^{5}x.**

**Sol.**

[put sin x = t â‡’ cos x dx = dt]

**Ex.21 Evaluate **

**Sol.**

**Ex.22 Integrate 1/(sin ^{3} x cos^{5}x).**

**Sol.** Here the integrand is sin^{â€“3} x cos^{â€“5}x. It is of type sin^{m} x cos^{n} x,where m + n = â€“3 â€“5 = â€“8 i.e., â€“ve even integer

Now put tan x = t so that sec^{2 }x 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 sec^{2}x dx = dt

**Ex.24 Evaluate **

**Sol.**

Put x â€“ Î² = y â‡’ dx = dy

Given integral

Now put sinÎ¸ + cosÎ¸ tan y = z^{2} â‡’ cosq sec^{2} y dy = 2z dz

**Ex.25 Evaluate ****dx**

**Sol.**

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