Integrals

Formulas for Integrals

Power Rule for Integration

$\int x^n \,dx = \frac{x^{n+1}}{n+1} + C$ Holds for any real number n ≠ -1.

Integration by Parts

$\int u \,dv = uv - \int v \,du$ u and v are functions of x. The choice of u often follows the LIATE/ILATE rule.

Standard Integral (Inverse Tan)

$\int \frac{1}{x^2 + a^2} \,dx = \frac{1}{a} \tan^{-1}\left(\frac{x}{a}\right) + C$ a is a constant.

Standard Integral (Log)

$\int \frac{1}{x^2 - a^2} \,dx = \frac{1}{2a} \log\left|\frac{x-a}{x+a}\right| + C$ a is a constant.

Standard Integral (Inverse Sin)

$\int \frac{1}{\sqrt{a^2 - x^2}} \,dx = \sin^{-1}\left(\frac{x}{a}\right) + C$ a is a constant.

Fundamental Theorem of Calculus

$\int_a^b f(x) \,dx = F(b) - F(a)$ F(x) is the antiderivative of f(x), i.e., F'(x) = f(x).

Property of Definite Integrals (King Property)

$\int_a^b f(x) \,dx = \int_a^b f(a+b-x) \,dx$ Extremely useful for simplifying integrands in definite integrals.

Property for Even and Odd Functions

$\int_{-a}^{a} f(x) \,dx = 2\int_{0}^{a} f(x) \,dx$ This holds if f(x) is an even function. The integral is 0 if f(x) is an odd function.

Leibniz Rule for Differentiation Under the Integral Sign

$\frac{d}{dx} \int_{u(x)}^{v(x)} f(t) \,dt = f(v(x)) \cdot v'(x) - f(u(x)) \cdot u'(x)$ Used to find the derivative of a definite integral with variable limits.

Area Between Two Curves

$A = \int_a^b [f(x) - g(x)] \,dx$ Calculates the area where the curve y=f(x) is above y=g(x) from x=a to x=b.