How do you integrate by parts?
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.
Related Maths A Level answers
All answers ▸Find the integral of the following equation: y = cos^2(x)
The curve C is defined by x^3 – (4x^2 )y = 2y^3 – 3x – 2. Find the value of dy/dx at the point (3, 1).
a) Solve the following equation by completing the square: x^(2)+ 6x + 1= 0. b) Solve the following equation by factorisation: x^(2) - 4x - 5 = 0 c) Solve the following quadratic inequality: x^(2) - 4x - 5 < 0 (hint use your answer to part b)
Earth is being added to a pile so that, when the height of the pile is h metres, its volume is V cubic metres, where V = (h6 + 16) 1 2 − 4.Find the value of dV/dh when h = 2.