定积分与广义积分计算器

该计算器将尝试计算定积分（即带上下限的积分），包括广义积分，并显示求解步骤。

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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: $$$\int_{0}^{\pi}\left( 7 x^{2} \sin{\left(x \right)} \right)dx=-28 + 7 \pi^{2}\approx 41.0872308076255$$$

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定积分与广义积分计算器

逐步计算定积分和广义积分

Solution