Question

Find the function f and the value of the constant a such that: 2 ∫ f(t)dt x a = 2 cos x − 1

Answer 1

The function is
`-\sin x` and the constant of integration is
`C = - 1`.

Step-by-step explanation:

The resultant expression is equal to the sum of a constant multiplied by the integral of a given function and an integration constant. That is:

`a = k\cdot \int\limits {f(x)} \, dx + C`

Where:

`k` - Constant, dimensionless.

`C` - Integration constant, dimensionless.

By comparing terms,
`k = 2`,
`C = -1` and
`\int {f(x)} \, dx = \cos x`. Then,
`f(x)` is determined by deriving the cosine function:

`f(x) = (d)/(dx) (\cos x)`

`f(x) = -\sin x`

The function is
`-\sin x` and the constant of integration is
`C = - 1`.