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Mathematics

The Intuition of Integration by Parts

We explore integration by parts not as a memorized formula, but as a visual transformation of a rectangular area, derived directly from the product rule of differentiation.

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The Product Rule
u(x)v(x)
ddx(uv)=udvdx+vdudx\frac{d}{dx}(uv) = u \frac{dv}{dx} + v \frac{du}{dx}
Integration by parts is essentially the product rule of calculus in reverse. If we look at the derivative of a product of two functions, u and v, we get two distinct growth terms.
Step-by-step solver
1

Choose u and dv

Select u to be the part that becomes simpler when differentiated, and dv as the part that is easy to integrate.

u=x,dv=exdxu = x, dv = e^x dx
2

Differentiate u and integrate dv

Find du by differentiating x and v by integrating e^x.

du=dx,v=exdu = dx, v = e^x
3

Apply the formula

Plug your values into the integration by parts formula.

∫xexdx=(x)(ex)−∫exdx\int x e^x dx = (x)(e^x) - \int e^x dx
4

Finish the integration

Solve the final remaining integral and add the constant of integration.

xex−ex+Cx e^x - e^x + C

Original question

Explain integration by parts with a worked example, and include a diagram where it helps.

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