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Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE PDF Download

1. 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 & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE f(g(x))g'(x) dx = F(g(x)) + C.

If u = g(x), then du = g'(x) and Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE 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 & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

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 & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE then the substitution u = Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE may be effective.


Some Standard Substitutions
Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE


2. Integration Using Trigonometric Identities

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:

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

All these identities simplify integrand, that can be easily found out.


Solved Examples:

Ex.1 Evaluate Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE (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 & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Ex.2 Evaluate  Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE
Sol.  Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Ex.3 Evaluate Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE
Sol.

Let u = x+ 2   ⇒   du = 4x3 dx

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Ex.4 Evaluate Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Sol.

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

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE
Ex.5 Evaluate  Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Sol. Let u  =  Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE . Then u= x + 4, so x = u–4 and dx = 2u du.

Therefore  Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE


Ex.6 Evaluate  Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Sol. Rewrite the integrand as follows :

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE = – ln (e-x + 1) + c   (∴  e-x + 1 > 0)


Ex.7 Evaluate  Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEEsec x dx

Sol. Multiply the integrand sec x by sec x + tan x and divide by the same quantity :
Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

 

Ex.8 Evaluate Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEEcos x (4 - sin2 x) dx
Sol. Put sin x = t so that cos x dx = dt. Then the given integral = Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE


Ex.9 Integrate 

(i) Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE
(ii) Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE


Sol.

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE


Ex.10 Integrate 

(i) Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE
(ii) Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE


Sol.

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE


Ex.11 Integrate 

(i) Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE
(ii) Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE


Sol.

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

 

Ex.12 Integrate Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE


Sol.

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Ex.13 Integrate cos5x.

Sol.

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE [put sin x = t ⇒ cos x dx = dt] 

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Ex.14 Evaluate Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Sol.

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Ex.15 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 & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

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

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Ex.16 Integrate Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE


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 & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE ,putting tan x = t and sec2x dx = dt

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Ex.17 Evaluate  Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Sol.

Put x – β = y  ⇒  dx = dy

Given integral 

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

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

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

 

Ex.18 Evaluate Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEEdx

Sol.

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE

The document Integration by Substitution & Trigonometric Identities | Mathematics (Maths) Class 12 - JEE is a part of the JEE Course Mathematics (Maths) Class 12.
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FAQs on Integration by Substitution & Trigonometric Identities - Mathematics (Maths) Class 12 - JEE

1. What is integration by substitution?
Ans. Integration by substitution is a technique used in calculus to simplify integrals by making a substitution of variables. It involves rewriting an integral in terms of a new variable, such that the integral becomes easier to evaluate.
2. How do you perform integration by substitution?
Ans. To perform integration by substitution, follow these steps: 1. Identify a part of the integrand that can be substituted with a new variable. 2. Determine the derivative of the new variable. 3. Rewrite the integral using the new variable. 4. Evaluate the integral using the new variable. 5. Replace the new variable with the original variable in the final answer.
3. What are trigonometric identities used for in integration?
Ans. Trigonometric identities are mathematical equations involving trigonometric functions that are useful in simplifying integrals involving trigonometric functions. By using trigonometric identities, we can transform complicated integrals into simpler forms, making them easier to evaluate.
4. How can trigonometric identities be applied in integration?
Ans. Trigonometric identities can be applied in integration by substituting certain trigonometric functions with their equivalent forms using the identities. This substitution helps simplify the integral, often leading to a more manageable form that can be easily evaluated.
5. Can integration by substitution and trigonometric identities be used together?
Ans. Yes, integration by substitution and trigonometric identities can be used together in certain cases. Depending on the complexity of the integral, using a combination of both techniques may be necessary to simplify the integrand and evaluate the integral effectively.
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