Indefinite Integration

Indefinite Integration (antidifferentiation) is the reverse process of differentiation. Given a function f(x), we seek a function F(x) such that F'(x) = f(x). The integral of f(x) with respect to x is written as integral f(x)dx = F(x) + C, where C is the constant of integration.

Standard Integrals:

• integral x^n dx = x^(n+1)/(n+1) + C, for n != -1
• integral 1/x dx = ln|x| + C
• integral e^x dx = e^x + C
• integral a^x dx = a^x/ln(a) + C
• integral sin(x) dx = -cos(x) + C
• integral cos(x) dx = sin(x) + C
• integral $sec^{2}$(x) dx = tan(x) + C
• integral $csc^{2}$(x) dx = -cot(x) + C
• integral sec(x)tan(x) dx = sec(x) + C
• integral csc(x)cot(x) dx = -csc(x) + C
• integral 1/$\sqrt{1-x^{2}}$ dx = arcsin(x) + C
• integral -1/$\sqrt{1-x^{2}}$ dx = arccos(x) + C
• integral 1/(1+$x^{2}$) dx = arctan(x) + C
• integral 1/(x*$\sqrt{x^{2}-1}$) dx = arcsec(x) + C

Method 1: Substitution (Change of Variable) If the integrand contains a composite function f(g(x))*g'(x), substitute u = g(x), du = g'(x)dx: integral f(g(x))*g'(x)dx = integral f(u)du

Method 2: Integration by Parts integral u dv = uv - integral v du Choice of u follows LIATE order: Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential.

Method 3: Partial Fractions For rational functions P(x)/Q(x) where degree P < degree Q:

• Linear factor (ax+b): A/(ax+b)
• Repeated linear factor (ax+b)^n: A₁/(ax+b) + A₂/(ax+b)² + ... + An/(ax+b)^n
• Irreducible quadratic ($ax^{2}$+bx+c): (Ax+B)/($ax^{2}$+bx+c)

Method 4: Trigonometric Integrals

• Powers of sin and cos: use reduction or half-angle identities
• sin(mx)cos(nx): use product-to-sum formulas
• integral sin^m(x)cos^n(x)dx: if m or n is odd, substitute the even-powered function

Method 5: Trigonometric Substitution

• $\sqrt{a^{2}-x^{2}}$: let x = a*sin($\theta$)
• $\sqrt{a^{2}+x^{2}}$: let x = a*tan($\theta$)
• $\sqrt{x^{2}-a^{2}}$: let x = a*sec($\theta$)

Method 6: Special Forms

• integral e^x[f(x)+f'(x)]dx = e^x*f(x) + C
• integral 1/($x^{2}$+$a^{2}$)dx = (1/a)arctan(x/a) + C
• integral 1/($x^{2}$-$a^{2}$)dx = (1/2a)ln|(x-a)/(x+a)| + C
• integral 1/$\sqrt{x^{2}+a^{2}}$dx = ln|x+$\sqrt{x^{2}+a^{2}}$| + C
• integral 1/$\sqrt{x^{2}-a^{2}}$dx = ln|x+$\sqrt{x^{2}-a^{2}}$| + C
• integral $\sqrt{a^{2}-x^{2}}$dx = (x/2)$\sqrt{a^{2}-x^{2}}$ + ($\frac{a^{2}}{2}$)arcsin(x/a) + C