To solve the integral [tex]\(\int_0^{\pi / 3} \cos (x) \sin (\sin (x)) \, dx\)[/tex], follow the steps below:
1. Identify the integral to be evaluated:
[tex]\[
\int_0^{\pi / 3} \cos (x) \sin (\sin (x)) \, dx
\][/tex]
2. Consider using a substitution method:
Let’s set [tex]\( u = \sin(x) \)[/tex]. Then, we need to find [tex]\( du \)[/tex] to replace [tex]\( dx \)[/tex] in the integral.
3. Compute the derivative [tex]\( du \)[/tex]:
[tex]\[
du = \cos(x) \, dx
\][/tex]
This means [tex]\( dx = \frac{du}{\cos(x)} \)[/tex].
4. Change the limits of integration according to the substitution [tex]\( u = \sin(x) \)[/tex]:
When [tex]\( x \)[/tex] changes from [tex]\( 0 \)[/tex] to [tex]\( \frac{\pi}{3} \)[/tex]:
- When [tex]\( x = 0 \)[/tex], [tex]\( u = \sin(0) = 0 \)[/tex].
- When [tex]\( x = \frac{\pi}{3} \)[/tex], [tex]\( u = \sin\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{2} \)[/tex].
5. Rewrite the integral using the new variable [tex]\( u \)[/tex]:
[tex]\[
\int_0^{\pi / 3} \cos (x) \sin (\sin (x)) \, dx = \int_0^{\frac{\sqrt{3}}{2}} \sin(u) \, du
\][/tex]
Since [tex]\( du = \cos(x) \, dx \)[/tex], the [tex]\( \cos(x) \)[/tex] term cancels out:
[tex]\[
\int \sin(u) \, du
\][/tex]
6. Integrate [tex]\( \sin(u) \)[/tex]:
The integral of [tex]\( \sin(u) \)[/tex] with respect to [tex]\( u \)[/tex] is:
[tex]\[
\int \sin(u) \, du = -\cos(u)
\][/tex]
7. Apply the limits of integration:
[tex]\[
\left[ -\cos(u) \right]_0^{\frac{\sqrt{3}}{2}}
\][/tex]
8. Evaluate at the upper and lower limits:
[tex]\[
-\cos\left(\frac{\sqrt{3}}{2}\right) - \left( -\cos(0) \right)
\][/tex]
Since [tex]\( \cos(0) = 1 \)[/tex]:
[tex]\[
-\cos\left( \frac{\sqrt{3}}{2} \right) + 1
\][/tex]
9. Simplify the expression:
Evaluating [tex]\( \cos\left( \frac{\sqrt{3}}{2} \right) \)[/tex] directly can be challenging without a numerical method, but the integral evaluates numerically to be approximately [tex]\( 0.3521406551475431 \)[/tex].
Therefore, the value of the integral
[tex]\[
\int_0^{\pi / 3} \cos (x) \sin (\sin (x)) \, dx
\][/tex]
is approximately [tex]\( 0.3521406551475431 \)[/tex].
