Here are some common mathematical equations:

Pythagorean Theorem: a^2 + b^2 = c^2

Quadratic Formula: x = (-b ± √(b^2 – 4ac)) / 2a

Distance Formula: d = √((x2 – x1)^2 + (y2 – y1)^2)

Slope-Intercept Form: y = mx + b

Fundamental Theorem of Calculus: ∫(a to b) f(x)dx = F(b) – F(a), where F(x) is the antiderivative of f(x)

Euler’s Formula: e^(ix) = cos(x) + i sin(x)

Law of Cosines: c^2 = a^2 + b^2 – 2abcos(C)

Law of Sines: a/sin(A) = b/sin(B) = c/sin(C)

Derivative of a Function: f'(x) = lim(h → 0) [f(x+h) – f(x)] / h

Integration by Parts: ∫ u dv = uv – ∫ v du

Note that this is not an exhaustive list and there are many other important mathematical equations out there.

Rajesh Sharma

integration by parts theorem with proof

The integration by parts formula is used to integrate the product of two functions. It states that:

∫u dv = uv – ∫v du

where u and v are functions of x, and du/dx and dv/dx are their derivatives with respect to x.

To prove this formula, we start with the product rule:

(d/dx)(uv) = u dv/dx + v du/dx

Rearranging this equation, we get:

u dv/dx = (d/dx)(uv) – v du/dx

Multiplying both sides by dx and integrating, we get:

∫u dv = ∫[(d/dx)(uv) – v du/dx] dx

Using the integral of a derivative rule, we can simplify the