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

Answer:

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

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:

$f(x) = \frac{d}{dx} (\cos x)$

$f(x) = -\sin x$

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