Definite & Indefinite Integrals Formulas

All key formulas grouped by subtopic. Each one has a quick reminder and common mistakes to watch for.

10 formulas · 8 subtopics

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1

💡 For n = -1: ∫(1/x)dx = ln|x| + C. Always add the constant of integration for indefinite integrals.

\int f(g(x)) \cdot g'(x) \, dx = \int f(t) \, dt, \quad t = g(x)

💡 Choose substitution to simplify the integrand. Don't forget to convert dx to dt and change limits for definite integrals.

\int u \, dv = uv - \int v \, du

💡 ILATE rule for choosing u: Inverse trig > Log > Algebraic > Trig > Exponential.

⚠ Forgetting to apply by parts again when the new integral is still a product (sometimes needed twice).

\int_a^b f(x) \, dx = \int_a^b f(a+b-x) \, dx

💡 Most powerful property for definite integrals. Use when f(x) + f(a+b-x) simplifies nicely.

\int_{-a}^{a} f(x) \, dx = \begin{cases} 2\int_0^a f(x)\,dx & \text{if } f(-x)=f(x) \\ 0 & \text{if } f(-x)=-f(x) \end{cases}

💡 Check parity first for symmetric limits. Saves huge computation for odd functions (integral = 0).

\int_0^{nT} f(x) \, dx = n \int_0^T f(x) \, dx \text{ if } f(x+T)=f(x)

💡 For sin²x, cos²x: period = π. For |sinx|, |cosx|: period = π.

\int_0^{\pi/2} \sin^n x \, dx = \int_0^{\pi/2} \cos^n x \, dx = \frac{(n-1)!!}{n!!} \cdot k

💡 k = π/2 if n is even, k = 1 if n is odd. Double factorial: n!! = n(n-2)(n-4)...

\frac{P(x)}{(x-a)(x-b)} = \frac{A}{x-a} + \frac{B}{x-b}

💡 For repeated roots: A/(x-a) + B/(x-a)². For irreducible quadratic: (Ax+B)/(x²+px+q).

\int_0^1 f(x)\,dx = \lim_{n\to\infty} \frac{1}{n}\sum_{r=1}^{n} f\left(\frac{r}{n}\right)

💡 Replace r/n → x, 1/n → dx. Limits: r=1 gives x=0, r=n gives x=1.

#10

\beta(m,n) = \int_0^1 x^{m-1}(1-x)^{n-1}\,dx = \frac{\Gamma(m)\Gamma(n)}{\Gamma(m+n)}

💡 Useful for integrals of the form ∫₀¹ xᵃ(1-x)ᵇ dx. Γ(n) = (n-1)! for positive integers.