Hello,
I didn't succeed in solving the exercise below.
2.1) Solve the following:
. . . . .\(\displaystyle \displaystyle{ \sum_{n\,=\,1}^{\infty} \,}\) \(\displaystyle \sqrt[3]{n^2\,}\, \cdot\, \) \(\displaystyle \displaystyle{ \int_{0}^{\frac{1}{n^2}}\,}\) \(\displaystyle \dfrac{\sin^{10}(x)}{3\, +\, \cos^5(x)}\, dx\)
by using the following:
. . . . .\(\displaystyle \mbox{If }\, m\, =\, \min_{[a,\,b]}\,f(x)\, \mbox{ and }\, M\, =\, \max_{[a,\,b]}\, f(x),\, \mbox{ then:}\)
. . . . .\(\displaystyle m\, \cdot\, (b\, -\, a)\, \leq\, \) \(\displaystyle \displaystyle{ \int_a^b \,}\) \(\displaystyle f(x)\, dx\, \leq\, M\, \cdot\, (b\, -\, a)\)
I think that I need to show here that the function in the integral is monotonous in its' domain of definition, but I don't know how.
Can you show me how to solve this?
I didn't succeed in solving the exercise below.
2.1) Solve the following:
. . . . .\(\displaystyle \displaystyle{ \sum_{n\,=\,1}^{\infty} \,}\) \(\displaystyle \sqrt[3]{n^2\,}\, \cdot\, \) \(\displaystyle \displaystyle{ \int_{0}^{\frac{1}{n^2}}\,}\) \(\displaystyle \dfrac{\sin^{10}(x)}{3\, +\, \cos^5(x)}\, dx\)
by using the following:
. . . . .\(\displaystyle \mbox{If }\, m\, =\, \min_{[a,\,b]}\,f(x)\, \mbox{ and }\, M\, =\, \max_{[a,\,b]}\, f(x),\, \mbox{ then:}\)
. . . . .\(\displaystyle m\, \cdot\, (b\, -\, a)\, \leq\, \) \(\displaystyle \displaystyle{ \int_a^b \,}\) \(\displaystyle f(x)\, dx\, \leq\, M\, \cdot\, (b\, -\, a)\)
I think that I need to show here that the function in the integral is monotonous in its' domain of definition, but I don't know how.
Can you show me how to solve this?
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