Määrättyjen ja epäoleellisten integraalien laskin
Laske määrättyjä ja epäoleellisia integraaleja askel askeleelta
Laskin yrittää laskea määrätyn (eli rajoilla varustetun) integraalin, mukaan lukien epäoleelliset tapaukset, ja näyttää välivaiheet.
Solution
Your input: calculate $$$\int_{- \frac{\pi}{6}}^{\frac{\pi}{2}}\left( 144 \sin^{2}{\left(\theta \right)} \right)d\theta$$$
First, calculate the corresponding indefinite integral: $$$\int{144 \sin^{2}{\left(\theta \right)} d \theta}=72 \theta - 36 \sin{\left(2 \theta \right)}$$$ (for steps, see indefinite integral calculator)
According to the Fundamental Theorem of Calculus, $$$\int_a^b F(x) dx=f(b)-f(a)$$$, so just evaluate the integral at the endpoints, and that's the answer.
$$$\left(72 \theta - 36 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=\frac{\pi}{2}\right)}=36 \pi$$$
$$$\left(72 \theta - 36 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=- \frac{\pi}{6}\right)}=- 12 \pi + 18 \sqrt{3}$$$
$$$\int_{- \frac{\pi}{6}}^{\frac{\pi}{2}}\left( 144 \sin^{2}{\left(\theta \right)} \right)d\theta=\left(72 \theta - 36 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=\frac{\pi}{2}\right)}-\left(72 \theta - 36 \sin{\left(2 \theta \right)}\right)|_{\left(\theta=- \frac{\pi}{6}\right)}=- 18 \sqrt{3} + 48 \pi$$$
Answer: $$$\int_{- \frac{\pi}{6}}^{\frac{\pi}{2}}\left( 144 \sin^{2}{\left(\theta \right)} \right)d\theta=- 18 \sqrt{3} + 48 \pi\approx 119.61953283607$$$