Integration of Irrational Functions

 Table of contents Definition Solved Examples Integrals of the type   where x & y are linear or quadratic expressions Integration Of A Binomial Differential

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 Rthen 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  then rationalization of the integral is effected by the substitution  where m has the same sense as above.

Solved Examples

Ex.1 Evaluate

Sol.

Rationalizing the denominator, we have

Ex.2 Evaluate I =

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

x = t6, dx = 6t5 dt.

Ex.3 Evaluate I =

Sol. The integrand is a rational function of  therefore we put 2x – 3 = t6, whence

Returning to x, we get

Ex.4 Evaluate

Sol.

Let x = t3 ⇒ dx = 3t2 then

Ex.5 Evaluate I =

Sol. The integrand is a rational function of x and the expression   therefore let us introduce the substitution

Integrals of the type where x & y are linear or quadratic expressions

Ex.1 Integrate

Sol.

Put 4x + 3 = t2, so that 4dx = 2tdt and (2x + 1)

Ex.2 Evaluate

Sol.

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

∴

dividing the numerator by the denominator

Ex.3 Integrate

Sol.

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

Ex.4 Integrate

Sol.

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

Also x = (1/t) – 1.

Ex.5 Evaluate

Sol.

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

∴

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

[∵ t = 1/x]

Ex.6 Evaluate I =

Sol.

Integration Of A Binomial Differential

The integral  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 : is an integer. We put a + bxn = tα, where α is the denominator of the fraction p.

Case III :+ p is an integer we put a + bxn = tαxn, where a is the denominator of the fraction p.

Ex.1 Evaluate I =

Sol.

Here p = 2, i.e. an integer, hence we have case I.

Ex.2 Evaluate I =

Sol.

i.e. an integer.we have case II. Let us make the substitution. Hence ,

Ex.3 Evaluate I =

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

Substituting these expression into the integral, we obtain

Returning to x, we get I =

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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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.

Mathematics (Maths) for JEE Main & Advanced

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