## Staffy 2 years ago Evaluate the following integrals: sin^2(x)dx from 0 to 2π, |sin x| dx from 0 to 2π, (x^n)(e^−x)dx from 0 to ∞ sin(ax)sin(bx)dx from 0 to π Consider all possible cases (n is a positive integer)

1) hint : sin^2 x = (1-cos2x)/2

2) divide 2 case for : Area1 = int (sinx) dx [0,pi] area2 = int(-sinx)dx [pi,2pi] the total area is .........

3. Staffy

Why is it negative sinx?

because the absolut function must be positive number, so the area of under x-axes should give a (-) sign too, so that (-)(-) be postive

5. Staffy

That makes sense, thanks

let's look this graph |dw:1355024687100:dw|

7. UnkleRhaukus

4) \begin{align*} \int\limits_{0}^\pi\sin(mx)\sin(nx)\text dx &=\frac12\int\limits_{0}^\pi\cos((m-n)x)-\cos((m+n)x)\text dx\\ \\ &=\frac12\left(\frac{\sin((m-n)x)}{m-n}-\frac{\sin((m+n)x)}{m+n}\Big|_0^\pi\right)\\ &=\dots\\ &=\dots\\ \\ \end{align*}