Question

How do you find `\int _ { 0} ^ { 1} x ^ { 5} + x ^ { 3} d x`?

Answer 1

The integral equals `5/12`

Explanation:

Start by separating the integrals, it will be easier to work with.

`I = int_0^1 x^5dx + int_0^1 x^3dx`

Use the power rule for integration, which states that `int(x^n)dx = x^(n + 1)/(n + 1) + C`.

`I = [1/6x^6]_0^1 + [1/4x^4]_0^1`

Now use the fundamental theorem of calculus to evaluate, which states that `int_a^b f(x) = F(b) - F(a)`, where `F'(x) = f(x)` and `f(x)` is a continuous function on `[a, b]`.

`I = 1/6 - 0 + 1/4 - 0`

`I = 5/12`

Hopefully this helps!