This is one of the trickier methods of integration, and it requires some practise. The basic idea is to split a function which would be difficult to integrate into two parts. Differentiating one part and integrating the other will then lead to a function which is much easier to integrate.

The formula is that the integral of u dv = uv - the integral of v du. It is best demonstrated with an example:

Let's integrate f(x) = xcos(x)

We can see that x will disppear if we differentiate it, so let's set x = u and cos(x) = dv.

Differentiating u and integrating dv then gives du = 1 and v = sin(x)

Now we substitute these into the formula: xsin(x) - integral of sin(x)

Sin(x) is easy to integrate, it is just -cos(x). Now we have our answer! The integral of xcos(x) = xsin(x) + cos(x) + c, where c is our unknown (and always necessary!) constant of integration.

Solve the

Related Maths A Level answers

Find the area between the curves C_1, C_2 and the lines x=0 and x=1, where C_1 is the curve y = x^2 and C_2 is the curve y = x^3.

How do you integrate by parts?

The curve C has the equation: 2(x^2)y + 2x + 4y – cos (πy) = 17 use implicit differentiation to find dy/dx in terms of x and y

Differentiate y = x^3 +x^2 - 4x +5 with respects to x.

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