Calculadora de Integrais Definidas e Impróprias

A calculadora tentará calcular a integral definida (isto é, com limites), inclusive no caso impróprio, com os passos mostrados.

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If you need `-oo`, type -inf.

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Solution

Your input: calculate $$$\int_{0}^{\frac{\pi}{4}}\left( \sin^{2}{\left(2 x \right)} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\sin^{2}{\left(2 x \right)} d x}=\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}$$$ (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.

$$$\left(\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}\right)|_{\left(x=\frac{\pi}{4}\right)}=\frac{\pi}{8}$$$

$$$\left(\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}\right)|_{\left(x=0\right)}=0$$$

$$$\int_{0}^{\frac{\pi}{4}}\left( \sin^{2}{\left(2 x \right)} \right)dx=\left(\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}\right)|_{\left(x=\frac{\pi}{4}\right)}-\left(\frac{x}{2} - \frac{\sin{\left(4 x \right)}}{8}\right)|_{\left(x=0\right)}=\frac{\pi}{8}$$$

Answer: $$$\int_{0}^{\frac{\pi}{4}}\left( \sin^{2}{\left(2 x \right)} \right)dx=\frac{\pi}{8}\approx 0.392699081698724$$$

Calculadora de Integrais Definidas e Impróprias

Calcule integrais definidas e impróprias passo a passo

Solution