Integration: Advanced Techniques & Reduction

This session covers advanced integration techniques that go beyond the standard methods of substitution and parts. These include integration of rational functions via partial fractions, integration of irrational functions, reduction formulas, and special techniques for trigonometric and exponential integrals. While individual questions on these advanced techniques are rare in JEE Main, they appear as components of larger problems in definite integration and differential equations.

Partial Fraction Decomposition:

Every proper rational function P(x)/Q(x) (deg P < deg Q) can be decomposed into simpler fractions:

1. Distinct linear factors: A/(x-a) + B/(x-b) + ...
2. Repeated linear factors: A/(x-a) + B/(x-a)² + C/(x-a)³ + ...
3. Irreducible quadratic factors: (Ax+B)/($x^{2}$+px+q) where $p^{2}$-4q < 0
4. Repeated quadratic factors: (Ax+B)/($x^{2}$+px+q) + (Cx+D)/($x^{2}$+px+q)² + ...

For improper fractions (deg P >= deg Q), perform polynomial long division first to get polynomial + proper fraction.

Integration of Irrational Functions:

1. Linear irrationals $\sqrt{ax+b}$: Substitute t = $\sqrt{ax+b}$ or $t^{2}$ = ax+b.
2. Quadratic irrationals $\sqrt{ax^{2}+bx+c}$: Complete the square, then use trigonometric or hyperbolic substitution.
   • $\sqrt{a^{2}-x^{2}}$: x = a sin($\theta$)
   • $\sqrt{a^{2}+x^{2}}$: x = a tan($\theta$)
   • $\sqrt{x^{2}-a^{2}}$: x = a sec($\theta$) 3.
     Euler substitutions: For $\sqrt{ax^{2}+bx+c}$ when a > 0: t = $\sqrt{a}$*x + $\sqrt{ax^{2}+bx+c}$.
     When c > 0: t = xt + $\sqrt{c}$. These rationalize the integral.

Reduction Formulas:

A reduction formula expresses $I_{n}$ in terms of I_(n-1), I_(n-2), or similar. Derived using integration by parts.

Key reduction formulas:

• $I_{n}$ = integral of sin^n(x) dx: $I_{n}$ = (-sin^(n-1)x * cosx)/n + (n-1)/n * I_(n-2)
• $J_{n}$ = integral of cos^n(x) dx: $J_{n}$ = (cos^(n-1)x * sinx)/n + (n-1)/n * J_(n-2)
• $K_{n}$ = integral of tan^n(x) dx: $K_{n}$ = tan^(n-1)x/(n-1) - K_(n-2)
• $L_{n}$ = integral of sec^n(x) dx: $L_{n}$ = sec^(n-2)x * tanx/(n-1) + (n-2)/(n-1) * L_(n-2)
• $M_{n}$ = integral of x^n * e^x dx: $M_{n}$ = x^n * e^x - n * M_(n-1)

Wallis' Formula (Definite Integral):

integral(0 to $\frac{\pi}{2}$) sin^n(x) dx = integral(0 to $\frac{\pi}{2}$) cos^n(x) dx:

• n even: [(n-1)(n-3)...3.1] / [n(n-2)...4.2] * $\frac{\pi}{2}$
• n odd: [(n-1)(n-3)...4.2] / [n(n-2)...5.3]

Special Integrals:

1. integral dx/($x^{2}$+$a^{2}$) = (1/a)arctan(x/a) + C
2. integral dx/$\sqrt{a^{2}-x^{2}}$ = arcsin(x/a) + C
3. integral dx/$\sqrt{x^{2}+a^{2}}$ = ln|x+$\sqrt{x^{2}+a^{2}}$| + C
4. integral dx/$\sqrt{x^{2}-a^{2}}$ = ln|x+$\sqrt{x^{2}-a^{2}}$| + C
5. integral $\sqrt{a^{2}-x^{2}}$ dx = (x/2)$\sqrt{a^{2}-x^{2}}$ + ($\frac{a^{2}}{2}$)arcsin(x/a) + C 6.
   integral $\sqrt{x^{2}+a^{2}}$ dx = (x/2)$\sqrt{x^{2}+a^{2}}$ + ($\frac{a^{2}}{2}$)ln|x+$\sqrt{x^{2}+a^{2}}$| + C

Integration Using Differentiation Under the Integral Sign (Leibniz):

For parametric integrals I(a) = integral f(x,a) dx, differentiate with respect to a to simplify, integrate, then recover the original.

Beta and Gamma Functions (JEE Advanced level):

B(m,n) = integral(0 to 1) x^(m-1)(1-x)^(n-1) dx = Gamma(m)Gamma(n)/Gamma(m+n) Gamma(n+1) = n! for positive integers. Gamma($\frac{1}{2}$) = $\sqrt{\pi}$.