Integration by Substitution & Trigonometric Identities

# 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  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.

## 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:

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

## Solved Examples:

Ex.1 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.2 Evaluate
Sol.

Ex.3 Evaluate
Sol.

Let u = x+ 2   ⇒   du = 4x3 dx

Ex.4 Evaluate

Sol.

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

Ex.5 Evaluate

Sol. Let u  =   . Then u= x + 4, so x = u–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 cos5x.

Sol.

[put sin x = t ⇒ cos x dx = dt]

Ex.14 Evaluate

Sol.

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

Now put tan x = t so that secx 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 sec2x dx = dt

Ex.17 Evaluate

Sol.

Put x – β = y  ⇒  dx = dy

Given integral

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

Ex.18 Evaluate dx

Sol.

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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## 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.

## Mathematics (Maths) Class 12

204 videos|288 docs|139 tests

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