Mathematics

Mathematics

Problem

Which of the following relations is correct for the integral of the product of two differentiable functions u(x)u(x) and v(x)v(x)? a) uv+∫v duuv + \displaystyle\int v\,du b) uv+Cuv + C c) uv−∫v duuv - \displaystyle\int v\,du d) uv+C\dfrac{u}{v} + C
Intuitive explanation

Integration by parts is about trading one integral for another by using the product rule in reverse. The key insight is that ddx[u(x)v(x)]=u′v+uv′\frac{d}{dx}[u(x)v(x)] = u'v + uv', so rearranging gives you a formula to convert ∫u dv\int u \, dv into something involving ∫v du\int v \, du instead—useful when one part is easier to integrate than the other.

Intuitive explanation

Integration by parts is about trading one integral for another by using the product rule in reverse. The key insight is that ddx[u(x)v(x)]=u′v+uv′\frac{d}{dx}[u(x)v(x)] = u'v + uv', so rearranging gives you a formula to convert ∫u dv\int u \, dv into something involving ∫v du\int v \, du instead—useful when one part is easier to integrate than the other.

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