定積分與廣義積分計算器

此計算器將嘗試計算定積分（即帶有上下限的積分），包含廣義積分，並顯示步驟。

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Integrate with respect to:

Enter a lower limit:

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

Enter an upper limit:

If you need `oo`, type inf.

Please write without any differentials such as `dx`, `dy` etc.

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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: $$$\int_{- \frac{\pi}{3}}^{\frac{\pi}{3}}\left( \frac{\sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} \right)dx=0$$$

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定積分與廣義積分計算器

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Solution