Question

∫_(π/4)^(π/3)▒〖sinxdx;〗​

Answer 1

Ответ:

$н\sqrt2-1/2$

Объяснение:

$\int\limits^{\pi/3}_{\pi/4 } {sin(x)} \, dx = -cos(x)|\limits^{\pi/3}_{\pi/4 }=-cos(\pi/3 )+cos(\pi/4)=н\sqrt2-1/2$

Answer 2

To find the integral of $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \sin(x) \, dx$, you can use the fundamental theorem of calculus. Here's the step-by-step solution:

1. Find the antiderivative of $\sin(x)$, which is $-\cos(x) + C$, where $C$ is the constant of integration.

2. Evaluate this antiderivative at the upper and lower bounds of the integral and subtract:

   $-\cos\left(\frac{\pi}{3}\right) - (-\cos\left(\frac{\pi}{4}\right))$

3. Now, calculate the values of $\cos\left(\frac{\pi}{3}\right)$ and $\cos\left(\frac{\pi}{4}\right)$:

   $\cos\left(\frac{\pi}{3}\right) = \frac{1}{2}$ $\cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2}$

4. Substituting these values into the expression:

   $-\frac{1}{2} + \frac{\sqrt{2}}{2}$

So, the value of the integral $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}} \sin(x) \, dx$ is:

$-\frac{1}{2} + \frac{\sqrt{2}}{2}$