Integral of x sin x from 0 to π/2, solved by parts
The question: Evaluate the integral of x sin x from 0 to π/2.
The integral of x sin x from 0 to π/2 equals 1.
Watch the integral of x sin x from 0 to π/2 worked out step by step with integration by parts: choosing u and dv, plugging in, the limits, and a check. The answer is 1.
Step-by-step transcript
The question
Let's work out the integral of x sine x from zero to pi over two, and see exactly how it's done.
What we know and the plan
Here's what we know. We're integrating x times sine x, from zero to pi over two. We need one exact number. Because it's a product of two functions, we'll use integration by parts. It trades this integral for an easier one.
Set it up
We start with the integral from zero to pi over two of x sine x. Pick u to be x, because x gets simpler when we differentiate it. That leaves d v as sine x d x. Differentiate u, and d u is just d x. Integrate d v. The integral of sine is minus cos x. Now the parts formula. It's u times v, minus the integral of v d u.
Plug it in
Put in u and v. That's minus x cos x, from zero to pi over two, minus the integral of minus cos x. Two minuses make a plus, so now we're adding the integral of cos x. The bracket is zero. Cos of pi over two is zero, and at zero the x makes it zero. So only this integral is left.
Finish it
The integral of cos x is sine x, from zero to pi over two. Top limit first: sine of pi over two, minus the bottom limit, sine of zero. Sine of pi over two is one, and sine of zero is zero. So the answer is exactly one.
What it means
Here's the picture. The shaded area under x sine x, from zero to pi over two, is exactly one.
The answer
So the integral equals one.
Checking the answer
To check, differentiate our antiderivative, sine x minus x cos x. We get x sine x back, so it's right.
The common mistake
A common mistake is choosing u as sine x. That makes the new integral harder, not easier.
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