Calculus Gymnasiematematik(high school math) matematik 4

Integration by parts

Integration by  parts can be regarded as the inverse to the product rule for differentiation. Suppose U(X) and V(x) are  two differentiable functions. According to the product rule

dU(x)V(x)/dx = U(x) dV(x)/dx + V(x)dU(x)/dx = U(x) dV(x)/dx+ V(x)dU(x)/dx

Integrating both sides of this equation and transposing terms, we obtain
∫U(x)dV(x)/dx dx = U(x)V(x) – ∫ V(x)dU/dx dx

This is the general formula for integration by parts.

In each application we break up the integrand into a product of two pieces U and V’:  where  V’ is easier to integrate.

With this method one can differentiate for example lnx.

Let U=lnx then dU/dx = 1/x and dU = dx/x.

and dV = dx → V=x.

∫lnxdx = x lnx – ∫x1/xdx = xlnx – x + C


Av mattelararen

Licentiate of Philosophy in atomic Physics
Master of Science in Physics


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