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Calculadora de Integrais Definidas e Impróprias
Calcule integrais definidas e impróprias passo a passo
A calculadora tentará calcular a integral definida (isto é, com limites), inclusive no caso impróprio, com os passos mostrados.
Solution
Your input: calculate $$$\int_{0}^{\pi}\left( 7 x^{2} \sin{\left(x \right)} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{7 x^{2} \sin{\left(x \right)} d x}=- 7 x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} + 14 \cos{\left(x \right)}$$$ (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.
$$$\left(- 7 x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} + 14 \cos{\left(x \right)}\right)|_{\left(x=\pi\right)}=-14 + 7 \pi^{2}$$$
$$$\left(- 7 x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} + 14 \cos{\left(x \right)}\right)|_{\left(x=0\right)}=14$$$
$$$\int_{0}^{\pi}\left( 7 x^{2} \sin{\left(x \right)} \right)dx=\left(- 7 x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} + 14 \cos{\left(x \right)}\right)|_{\left(x=\pi\right)}-\left(- 7 x^{2} \cos{\left(x \right)} + 14 x \sin{\left(x \right)} + 14 \cos{\left(x \right)}\right)|_{\left(x=0\right)}=-28 + 7 \pi^{2}$$$
Answer: $$$\int_{0}^{\pi}\left( 7 x^{2} \sin{\left(x \right)} \right)dx=-28 + 7 \pi^{2}\approx 41.0872308076255$$$