定积分与广义积分计算器
逐步计算定积分和广义积分
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。
Solution
Your input: calculate $$$\int_{0}^{\frac{\pi}{2}}\left( \cos^{4}{\left(x \right)} \right)dx$$$
First, calculate the corresponding indefinite integral: $$$\int{\cos^{4}{\left(x \right)} d x}=\frac{12 x + 8 \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32}$$$ (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.
$$$\left(\frac{12 x + 8 \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32}\right)|_{\left(x=\frac{\pi}{2}\right)}=\frac{3 \pi}{16}$$$
$$$\left(\frac{12 x + 8 \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32}\right)|_{\left(x=0\right)}=0$$$
$$$\int_{0}^{\frac{\pi}{2}}\left( \cos^{4}{\left(x \right)} \right)dx=\left(\frac{12 x + 8 \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32}\right)|_{\left(x=\frac{\pi}{2}\right)}-\left(\frac{12 x + 8 \sin{\left(2 x \right)} + \sin{\left(4 x \right)}}{32}\right)|_{\left(x=0\right)}=\frac{3 \pi}{16}$$$
Answer: $$$\int_{0}^{\frac{\pi}{2}}\left( \cos^{4}{\left(x \right)} \right)dx=\frac{3 \pi}{16}\approx 0.589048622548086$$$