정적분 및 가적분 계산기

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Solution

Your input: calculate $$$\int_{7}^{49}\left( \frac{1}{x \ln{\left(x \right)}^{9}} \right)dx$$$

First, calculate the corresponding indefinite integral: $$$\int{\frac{1}{x \ln{\left(x \right)}^{9}} d x}=- \frac{1}{8 \ln{\left(x \right)}^{8}}$$$ (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.

$$$\left(- \frac{1}{8 \ln{\left(x \right)}^{8}}\right)|_{\left(x=49\right)}=- \frac{1}{8 \ln{\left(49 \right)}^{8}}$$$

$$$\left(- \frac{1}{8 \ln{\left(x \right)}^{8}}\right)|_{\left(x=7\right)}=- \frac{1}{8 \ln{\left(7 \right)}^{8}}$$$

$$$\int_{7}^{49}\left( \frac{1}{x \ln{\left(x \right)}^{9}} \right)dx=\left(- \frac{1}{8 \ln{\left(x \right)}^{8}}\right)|_{\left(x=49\right)}-\left(- \frac{1}{8 \ln{\left(x \right)}^{8}}\right)|_{\left(x=7\right)}=- \frac{1}{8 \ln{\left(49 \right)}^{8}} + \frac{1}{8 \ln{\left(7 \right)}^{8}}$$$

Answer: $$$\int_{7}^{49}\left( \frac{1}{x \ln{\left(x \right)}^{9}} \right)dx=- \frac{1}{8 \ln{\left(49 \right)}^{8}} + \frac{1}{8 \ln{\left(7 \right)}^{8}}\approx 0.000605658703782548$$$

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Solution