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Calculadora de integrales definidas e impropias
Calcular integrales definidas e impropias paso a paso
La calculadora intentará evaluar la integral definida (es decir, con límites de integración), incluyendo las impropias, mostrando los pasos.
Solution
Your input: calculate $$$\int_{- \frac{\pi}{3}}^{\frac{\pi}{3}}\left( \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} d x}=\frac{1}{2 \cos^{2}{\left(x \right)}}$$$ (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.
$$$\left(\frac{1}{2 \cos^{2}{\left(x \right)}}\right)|_{\left(x=\frac{\pi}{3}\right)}=2$$$
$$$\left(\frac{1}{2 \cos^{2}{\left(x \right)}}\right)|_{\left(x=- \frac{\pi}{3}\right)}=2$$$
$$$\int_{- \frac{\pi}{3}}^{\frac{\pi}{3}}\left( \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} \right)dx=\left(\frac{1}{2 \cos^{2}{\left(x \right)}}\right)|_{\left(x=\frac{\pi}{3}\right)}-\left(\frac{1}{2 \cos^{2}{\left(x \right)}}\right)|_{\left(x=- \frac{\pi}{3}\right)}=0$$$
Answer: $$$\int_{- \frac{\pi}{3}}^{\frac{\pi}{3}}\left( \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} \right)dx=0$$$