Question

Using integration by parts, integrate the

following:
a
2/a​∫ 100( x/a​ )⋅sin( nπx/a​)dx
0

Answer 1

The integration of
`(2/a)\int_(0)^(a) 100(x/a) \cdot sin(n \pi x /a) dx` is
`(100/n\pi) \int_(0)^(a) cos(n\pi x/a) dx.`

To integrate the given expression using integration by parts, we need to apply the formula:


`\int u dv = uv - \int v du`

1. Choose u and dv:
Let u = 100(x/a) and dv = sin(nπx/a) dx.

2. Find du and v:
Differentiating u, we get du = 100/a dx.
Integrating dv, we get v = -a/nπ cos(nπx/a).

3. Apply the integration by parts formula:

`\int u dv = uv - \int v du`

Plugging in the values, we have:
∫ (2/a) ∫₀ₐ 100(x/a) sin(nπx/a) dx = [(100(x/a))(-a/nπ cos(nπx/a))]₀ₐ - ∫₀ₐ -(a/nπ cos(nπx/a))(100/a) dx

4. Simplify and evaluate the definite integral limits:
At x = a, the first term becomes zero.
At x = 0, the second term becomes (100/a)(a/nπ) = 100/nπ.

5. Evaluate the remaining integral:

`\int_(0)^(a) -(a/n\pi cos(n\pi x/a))(100/a) dx = (100/n\pi) \int_(0)^(a) cos(n\pi x/a) dx`
6. Evaluate the integral of cos(nπx/a) using the appropriate formula or techniques.