Question

Evaluate the integral, please help!!!


`\int_(0)^(\pi)(1+cosx) dx` ​

Answer 1

`\displaystyle \int\limits^\pi_0 {(1+\cos x)} \, dx=\pi`

Explanation:

`\displaystyle \int\limits^\pi_0 {(1+\cos x)} \, dx\\\\=x+\sin x\biggr|^(\pi)_0\\\\=[\pi+\sin\pi]-[0+\sin0]\\\\=\pi`

Remember your antiderivatives!

Answer 2

pi

Explanation:

First solve the integral

`\int\ {(1+ cos x} )\, dx`

`\int\ {1} \, dx +\int\ {cos x} \, dx`

`\int\ {1} \, dx = x` and
`\int\ {cos x} \, dx = sin x`

x + sin x

Now consider the limit from 0 to π

`\lim_(0 \to \\pi ) (x+sin x)`

(π +sin π) -(0 +sin 0)

sin π = 0 and sin 0 = 0

π-0

π