Q&A for How to Integrate

Integration is used to understand the cumulative effect of a mathematical formula. In physical terms, it's finding the area bounded in part by the graph of a quadratic equation.

There is no trick for integrating cos x. You just have to memorize that the antiderivative of cos x is sin x.

There is no antiderivative that can be written in terms of elementary functions, so we cannot use the techniques in this article to integrate this function. However, if we assume the bounds are from 0 to π, we can substitute u = π - x to get π*∫log(sin u)du from 0 to π/2, since log(sin u) is symmetric about π/2. The integral can then be evaluated using the Beta function (this requires understanding of the Gamma function and its series expansions - see that article for details). ∫log(sin x)dx from 0 to π/2 = -π/2*log 2. The value of the original integral is therefore -π^2/2*log 2.

Integrating both sides, with respect to x, the integration of zero is not zero (a very common mistake) but c.