Question

Question #a9af4

Answer 1

Well, you can recall a basic property of integrals, the integrals of a constant and `sinx`, and the portion of the Fundamental Theorem of Calculus about evaluation of definite integrals:

• The integral of a sum is the sum of the integrals: `int f(x) + g(x)dx = int f(x)dx + int g(x)dx`.
• `int Cdx = Cx`, plus some integration constant if for an indefinite integral.
• `int sinxdx = -cosx`, plus some integration constant if for an indefinite integral.
• For definite integrals, if `F(x)` is the antiderivative, then `int_a^b f(x)dx = |[F(x)]|_(a)^(b) = F(b) - F(a)`.

So, simplify this and evaluate:

`color(blue)(int_(0)^(pi) 2 + sinxdx)`

`= int_(0)^(pi) 2dx + int_(0)^(pi) sinxdx`

`= 2|[x]|_(0)^(pi) + |[-cosx]|_(0)^(pi)`

`= 2(pi - 0) + [(-cospi) - (-cos0)]`

`= 2pi + [-(-1) - (-1)]`

`= 2pi + 1 + 1`

`= color(blue)(2pi + 2)`