Integration of Irrational Functions JEE Notes | EduRev

Mathematics (Maths) Class 12

JEE : Integration of Irrational Functions JEE Notes | EduRev

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Integration of Irrational Functions

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.

(i) If the integrand is a rational function of fractional powers of an independent variable x, i.e. the function RIntegration of Irrational Functions JEE Notes | EduRevthen the integral can be rationalized by the substitution x = tm, where m is the least common multiple of the numbers q1, q2, ...., qk.

(ii) If the integrand is a rational function of x and fractional powers of a linear fractional function of the form Integration of Irrational Functions JEE Notes | EduRev then rationalization of the integral is effected by the substitution Integration of Irrational Functions JEE Notes | EduRev where m has the same sense as above.

 

Ex.68 Evaluate  Integration of Irrational Functions JEE Notes | EduRev

Sol.

Rationalizing the denominator, we have  Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

 

Ex.69 Evaluate I = Integration of Irrational Functions JEE Notes | EduRev

 

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 JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Ex.70 Evaluate I = Integration of Irrational Functions JEE Notes | EduRev


Sol. The integrand is a rational function of Integration of Irrational Functions JEE Notes | EduRev therefore we put 2x – 3 = t6, whence

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev  

Returning to x, we get
Integration of Irrational Functions JEE Notes | EduRev


Ex.71 Evaluate Integration of Irrational Functions JEE Notes | EduRev

Sol.

Let x = t3 ⇒ dx = 3t2 then

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Ex.72 Evaluate I = Integration of Irrational Functions JEE Notes | EduRev


Sol. The integrand is a rational function of x and the expression  Integration of Irrational Functions JEE Notes | EduRev therefore let us introduce the substitution

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

INTEGRAL OF THE TYPE Integration of Irrational Functions JEE Notes | EduRev  WHERE X AND Y ARE LINEAR OR QUADRATIC EXPRESSION

 

Ex.73 Integrate Integration of Irrational Functions JEE Notes | EduRev

Sol.

Put 4x + 3 = t2, so that 4dx = 2tdt and (2x + 1)  Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Ex.74 Evaluate  Integration of Irrational Functions JEE Notes | EduRev

Sol.

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

∴  Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev  dividing the numerator by the denominator

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Ex.75 Integrate  Integration of Irrational Functions JEE Notes | EduRev

Sol.

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

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Ex.76 Integrate Integration of Irrational Functions JEE Notes | EduRev

Sol.

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

Also x = (1/t) – 1.

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Ex.77 Evaluate  Integration of Irrational Functions JEE Notes | EduRev

Sol.

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

∴  Integration of Irrational Functions JEE Notes | EduRev   Integration of Irrational Functions JEE Notes | EduRev

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

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev [∵ t = 1/x]

Ex.78 Evaluate I = Integration of Irrational Functions JEE Notes | EduRev

Sol. 

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

 

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

 Integration Of A Binomial Differential

The integral Integration of Irrational Functions JEE Notes | EduRev 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 JEE Notes | EduRev is an integer. We put a + bxn = tα, where α is the denominator of the fraction p.


Case III :Integration of Irrational Functions JEE Notes | EduRev+ p is an integer we put a + bxn = tαxn, where a is the denominator of the fraction p.


Ex.79 Evaluate I = Integration of Irrational Functions JEE Notes | EduRev

Sol.

Integration of Irrational Functions JEE Notes | EduRev  Here p = 2, i.e. an integer, hence we have case I.

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev


Ex.80 Evaluate I = Integration of Irrational Functions JEE Notes | EduRev

Sol. Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

 i.e. an integer.we have case II. Let us make the substitution. Hence , Integration of Irrational Functions JEE Notes | EduRev

Ex.81 Evaluate I = Integration of Irrational Functions JEE Notes | EduRev

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 JEE Notes | EduRev

Substituting these expression into the integral, we obtain

Integration of Irrational Functions JEE Notes | EduRev

Integration of Irrational Functions JEE Notes | EduRev

Returning to x, we get I =  Integration of Irrational Functions JEE Notes | EduRev

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