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3) Problem: Find the value of the definite integral: \[ \int_{\ln 3}^{\ln 8} \frac{d x}{\sqrt{e^{x}+1}} \]

9.3.3.13 Definite Integrals

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(2) Problem: Find the value of the definite integral: \[ \int_{0}^{\sqrt{5}} \sqrt{15-3 \mathrm{x}^{2}} d x \]

9.3.3.15 Definite Integrals

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3) Problem: Find the area of the figure, formed by the graphs of the functions \( y=x^{2}, y=8-x^{2} \).

9.7.15 The area of a region

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Problem: There are three identical boxes. There are 2 white and 2 black balls in the \( 1^{\text {st }} \) box; there are 3 black balls in the \( 2^{\text {nd }} \) box and 1 black and 5 white balls in the \( 3^{\text {rd }} \) box. Someone, randomly choosing a box, takes out a random ball. What is the probability that the ball will be white?

15.6.13 Definition and properties of probability

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Problem: The workers operate three machines. Let's denote by \( A_{i} \) the events, that during the day the machine with number \( i \) will require attention, \( i=1,2,3 \). Express the following events through \( A_{i} \) : \( A \) - three machines have a defect during the day; \( B \)-at least one machine has a defect during the day; \( C \)-no machine has a defect during the day.

15.6.17 Definition and properties of probability

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Problem: Find the image of line \( L \) for the mapping \( w=f(z) . L \) is the segment, connecting the points \( 1, i \), where \( w=(1+i) z+1-i \).

10.5.45 Conformal mappings

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Problem: Prove that the number \( \sqrt{3}-\sqrt{2} \) is irrational.

12.2.43 Number theory

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Problem: Find all values of the parameter \( a \), for which the minimum value of the function \( f(x)=4 x^{2}-4 a x+a^{2}-2 a++2 \) on the segment \( x \in[0 ; 2] \) is equal to 3 .

17.20 USE problems

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