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Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced PDF Download

Definition

Certain types of integrals of algebraic irrational expressions can be reduced to integrals of rational functions by a appropriate change of the variable. Such transformation of an integral is called its rationalization.

  1. If the integrand is a rational function of fractional powers of an independent variable x, i.e. the function RIntegration of Irrational Functions | Mathematics (Maths) for JEE Main & Advancedthen the integral can be rationalized by the substitution x = tm, where m is the least common multiple of the numbers q1, q2, ...., qk.
  2. If the integrand is a rational function of x and fractional powers of a linear fractional function of the form Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced then rationalization of the integral is effected by the substitution Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced where m has the same sense as above.

Solved Examples

Ex.1 Evaluate  Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol.

Rationalizing the denominator, we have  Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

 

Ex.2 Evaluate I = Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

 

Sol. The least common multiple of the numbers 3 and 6 is 6, therefore we make the substitution

x = t6, dx = 6t5 dt.

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Ex.3 Evaluate I = Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced


Sol. The integrand is a rational function of Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced therefore we put 2x – 3 = t6, whence

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced  

Returning to x, we get
Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced


Ex.4 Evaluate Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol.

Let x = t3 ⇒ dx = 3t2 then

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Ex.5 Evaluate I = Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced


Sol. The integrand is a rational function of x and the expression  Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced therefore let us introduce the substitution

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integrals of the type Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced  where x & y are linear or quadratic expressions

Ex.1 Integrate Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol.

Put 4x + 3 = t2, so that 4dx = 2tdt and (2x + 1)  Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Ex.2 Evaluate  Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol.

Put (x + 2) = t2, so that dx = 2t dt, Also x = t– 2.

∴  Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced  dividing the numerator by the denominator

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Ex.3 Integrate  Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol.

Put (x + 1) = t2, so that dx = 2t dt. Also x = t2 – 1.

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Ex.4 Integrate Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol.

Put (1 + x) = 1/t, so that dx = – (1/t2) dx.

Also x = (1/t) – 1.

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Ex.5 Evaluate  Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol.

Put x = 1/t, so that dx = – (1/t2) dt.

∴  Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced   Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Now put 1 + t2 = z2 so that t dt = z dz. Then

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced [∵ t = 1/x]

Ex.6 Evaluate I = Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol. 

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

 

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

 Integration Of A Binomial Differential

The integral Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced where m, n, p are rational numbers, is expressed through elementary functions only in the following three cases :


Case I : p is an integer. Then, if p > 0, the integrand is expanded by the formula of the binomial; but if p < 0, then we put x = tk, where k is the common denominator of the fractions and n.


Case II :Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced is an integer. We put a + bxn = tα, where α is the denominator of the fraction p.


Case III :Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced+ p is an integer we put a + bxn = tαxn, where a is the denominator of the fraction p.


Ex.1 Evaluate I = Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol.

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced  Here p = 2, i.e. an integer, hence we have case I.

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced


Ex.2 Evaluate I = Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol. Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

 i.e. an integer.we have case II. Let us make the substitution. Hence , Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Ex.3 Evaluate I = Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Sol. 

Here p = – 1/2 is a fraction, m+1/2 = -5/2 also a fraction, but m+1/n + p/2 = -5/2 -1/2 = -3 is an integer, i.e. we have case III, we put 1 + x4 = x4/2,

Hence   Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Substituting these expression into the integral, we obtain

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

Returning to x, we get I =  Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced

The document Integration of Irrational Functions | Mathematics (Maths) for JEE Main & Advanced is a part of the JEE Course Mathematics (Maths) for JEE Main & Advanced.
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FAQs on Integration of Irrational Functions - Mathematics (Maths) for JEE Main & Advanced

1. What are binomial differentials?
Ans. Binomial differentials are integrals of the form ∫(ax + b)^n dx, where a, b, and n are constants and x is the variable. These integrals involve binomial expressions raised to a power and are solved using specific integration techniques.
2. How do you integrate a binomial differential?
Ans. To integrate a binomial differential, you can use the power rule of integration. First, increase the power of the binomial by 1 and divide by the new power. Then, multiply the result by the reciprocal of the coefficient of x in the binomial. Finally, add the constant of integration to obtain the solution.
3. What are some examples of integrating binomial differentials?
Ans. Examples of integrating binomial differentials include ∫(2x + 3)^4 dx, ∫(3x^2 - 2)^3 dx, and ∫(5x - 1)^2 dx. These integrals involve different powers and coefficients in the binomial expression, requiring the application of the power rule and integration techniques.
4. Are there any special cases when integrating binomial differentials?
Ans. Yes, there are special cases when integrating binomial differentials. If the power of the binomial is -1, the integral represents the natural logarithm function. Similarly, if the power of the binomial is -2, the integral represents the inverse trigonometric function.
5. Can binomial differentials involve irrational functions?
Ans. Yes, binomial differentials can involve irrational functions, such as square roots or cube roots. In such cases, additional techniques like substitution or trigonometric substitution may be required to simplify the integral and find the solution.
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