Math Snap

STEP 1

What is this asking?
We're given the value of a definite integral of $f(x)$ and need to find the value of a different definite integral involving $f(\cos(x))$.
Watch out!
Don't get tricked by the different variables and limits of integration!
We need to use substitution wisely here.

STEP 2

1. Substitution
2. Calculate the new limits of integration
3. Evaluate the integral

STEP 3

Let's tackle this integral using u-substitution!
We're going to substitute $u = \cos(x)$.
Why? Because we see $f(\cos(x))$ in our integral, and we know something about $\int f(x) dx$!

STEP 4

Now, if $u = \cos(x)$, then its derivative is $\frac{du}{dx} = -\sin(x)$.
This means $du = -\sin(x) dx$, or equivalently $-du = \sin(x) dx$.
This is perfect because we have a $\sin(x) dx$ in our integral!

STEP 5

Let's rewrite our integral using our substitution:
$$\int_{0}^{\pi} f(\cos(x)) \sin(x) dx = \int f(u) (-du) = -\int f(u) du$$ Don't forget, we still need to figure out our new limits of integration!

STEP 6

Our original limits of integration were $x = 0$ and $x = \pi$.
Since we substituted $u = \cos(x)$, we need to find the corresponding values of $u$.

STEP 7

When $x = 0$, we have $u = \cos(0) = \textbf{1}$.

STEP 8

When $x = \pi$, we have $u = \cos(\pi) = \textbf{-1}$.

STEP 9

So our new limits of integration are $u = 1$ and $u = -1$.
Let's plug these in:
$$-\int_{1}^{-1} f(u) du$$

STEP 10

Remember the property of definite integrals: $\int_{a}^{b} f(x) dx = -\int_{b}^{a} f(x) dx$.
We can flip the limits of integration and change the sign:
$$-\int_{1}^{-1} f(u) du = \int_{-1}^{1} f(u) du$$

STEP 11

Look at that!
We know that $\int_{-1}^{1} f(x) dx = \pi$.
Since the variable of integration is just a dummy variable, we also know that $\int_{-1}^{1} f(u) du = \pi$!

Therefore, $\int_{0}^{\pi} f(\cos(x)) \sin(x) dx = \pi$, which is answer choice C.