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Määrättyjen ja epäoleellisten integraalien laskin
Laske määrättyjä ja epäoleellisia integraaleja askel askeleelta
Laskin yrittää laskea määrätyn (eli rajoilla varustetun) integraalin, mukaan lukien epäoleelliset tapaukset, ja näyttää välivaiheet.
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: $$$\int_{7}^{49}\left( \frac{1}{x \ln{\left(x \right)}^{9}} \right)dx=- \frac{1}{8 \ln{\left(49 \right)}^{8}} + \frac{1}{8 \ln{\left(7 \right)}^{8}}\approx 0.000605658703782548$$$