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定積分・広義積分計算機
定積分と広義積分を手順を追って計算する
この計算機は、定積分（すなわち上下限のある積分）を、広義積分も含めて、解法手順を示しながら計算を試みます。
Solution
Your input: calculate $$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\left( e^{x} \cos{\left(x \right)} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{e^{x} \cos{\left(x \right)} d x}=\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}$$$ (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.
$$$\left(\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}\right)|_{\left(x=\frac{\pi}{2}\right)}=\frac{e^{\frac{\pi}{2}}}{2}$$$
$$$\left(\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}\right)|_{\left(x=- \frac{\pi}{2}\right)}=- \frac{1}{2 e^{\frac{\pi}{2}}}$$$
$$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\left( e^{x} \cos{\left(x \right)} \right)dx=\left(\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}\right)|_{\left(x=\frac{\pi}{2}\right)}-\left(\frac{\sqrt{2} e^{x} \sin{\left(x + \frac{\pi}{4} \right)}}{2}\right)|_{\left(x=- \frac{\pi}{2}\right)}=\frac{1}{2 e^{\frac{\pi}{2}}} + \frac{e^{\frac{\pi}{2}}}{2}$$$
Answer: $$$\int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\left( e^{x} \cos{\left(x \right)} \right)dx=\frac{1}{2 e^{\frac{\pi}{2}}} + \frac{e^{\frac{\pi}{2}}}{2}\approx 2.50917847865806$$$