Table of contents  
Definition  
Solved Examples  
Integrals of the type where x & y are linear or quadratic expressions  
Integration Of A Binomial Differential 
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.
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 = t^{6}, dx = 6t^{5} dt.
Ex.3 Evaluate I =
Sol. The integrand is a rational function of therefore we put 2x – 3 = t^{6}, whence
Returning to x, we get
Ex.4 Evaluate
Sol.
Let x = t^{3} ⇒ dx = 3t^{2} then
Ex.5 Evaluate I =
Sol. The integrand is a rational function of x and the expression therefore let us introduce the substitution
Ex.1 Integrate
Sol.
Put 4x + 3 = t^{2}, so that 4dx = 2tdt and (2x + 1)
Ex.2 Evaluate
Sol.
Put (x + 2) = t^{2}, so that dx = 2t dt, Also x = t^{2 }– 2.
∴
dividing the numerator by the denominator
Ex.3 Integrate
Sol.
Put (x + 1) = t^{2}, so that dx = 2t dt. Also x = t^{2} – 1.
Ex.4 Integrate
Sol.
Put (1 + x) = 1/t, so that dx = – (1/t^{2}) dx.
Also x = (1/t) – 1.
Ex.5 Evaluate
Sol.
Put x = 1/t, so that dx = – (1/t^{2}) dt.
∴
Now put 1 + t^{2} = z^{2} so that t dt = z dz. Then
[∵ t = 1/x]
Ex.6 Evaluate I =
Sol.
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 = t^{k}, where k is the common denominator of the fractions and n.
Case II : is an integer. We put a + bx^{n} = t^{α}, where α is the denominator of the fraction p.
Case III :+ p is an integer we put a + bx^{n} = t^{α}x^{n}, 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 + x^{4} = x^{4/2},
Hence
Substituting these expression into the integral, we obtain
Returning to x, we get I =
209 videos443 docs143 tests

1. What are binomial differentials? 
2. How do you integrate a binomial differential? 
3. What are some examples of integrating binomial differentials? 
4. Are there any special cases when integrating binomial differentials? 
5. Can binomial differentials involve irrational functions? 
209 videos443 docs143 tests


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