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Problem: Change the order of integration: \[ I=\int_{0}^{a} d y \int_{\frac{y^{2}}{2 a}}^{a-\sqrt{a^{2}-y^{2}}} f d x+\int_{0}^{a} d y \int_{a+\sqrt{a^{2}-y^{2}}}^{2 a} f d x+\int_{0}^{2 a} d y \int_{\frac{y^{2}}{2 a}}^{2 a} f d x . \]

9.1.31 Double integrals

1.29 $

Problem: Find the mass of the flat figure \( D \) with density \( \rho \) : \[ \rho(x, y)=|y|, \quad D:(x-3)^{2}+y^{2} \leq 1, \quad x \geq 3 . \]

9.1.32 Double integrals

1.29 $

Problem: Calculate the area of the region: \[ D: y=20-x^{2}, y=-8 x \text {. } \]

9.1.33 Double integrals

1.03 $

Problem: Make a replacement of variables in the double integral: \[ G:\left\{\begin{array}{l} y=x \\ x^{2}+y^{2}=9 \\ x y=4(x>0, y \geq x) \end{array},\left\{\begin{array}{l} u=x^{2}+y^{2}-9 \\ v=x y-4 \end{array} .\right.\right. \]

9.1.11 Double integrals

1.54 $

Problem: Determine the domain of existence and express the following integral in terms of the Euler integrals: \[ I=\int_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\left(\frac{\cos x+\sin x}{\cos x-\sin x}\right)^{\cos 2 \alpha} d x \]

9.2.20 Integrals depending on a parameter

6.43 $

Problem: Prove the equality: \[ I=\int_{0}^{\frac{\pi}{2}} \sin ^{\rho} x d x \cdot \int_{0}^{\frac{\pi}{2}} \frac{d x}{\sin ^{\rho} x}=\frac{\pi}{2 \rho} \tan \frac{\pi \rho}{2}, \quad 0<\rho<1 . \]

9.2.21 Integrals depending on a parameter

5.15 $

Problem: Using differentiation or integration with respect to the parameter, calculate the improper integral. \[ \int_{0}^{\infty} \frac{\tan ^{-1} r x}{x\left(1+x^{2}\right)} d x, \quad(r \geq 0) . \]

9.2.22 Integrals depending on a parameter

5.15 $

Problem: Find the Laplace transform of the periodic function: \[ f(t)=B|\sin \omega t| . \]

9.2.23 Integrals depending on a parameter

3.09 $

Problem: Determine the region of existence of the improper integral and express it in Euler integrals. \[ \int_{0}^{1} \ln ^{\mathrm{p}} \frac{1}{x} d x . \]

9.2.24 Integrals depending on a parameter

2.06 $

Problem: Prove the convergence of the integral and find its value. \[ \int_{0}^{1} \sin \left(\ln \frac{1}{x}\right) \cdot \frac{x^{b}-x^{a}}{\ln x} d x, \quad a>0, b>0 . \]

9.2.25 Integrals depending on a parameter

5.15 $

Problem: Using the Dirichlet/Frullani/Fresnel/Euler/Poisson/Laplace integrals, calculate the following integral: \[ I=\int_{0}^{+\infty} \frac{\alpha \sin x-\sin \alpha x}{x^{2}} d x, \quad \alpha>0 . \]

9.2.26 Integrals depending on a parameter

3.86 $

Problem: Investigate the uniform convergence of the integral on the set \( Y \). \[ \int_{0}^{\infty} \frac{x d x}{1+(x-\alpha)^{4}}, \quad Y=(-\infty ; b], \quad b>0 . \]

9.2.28 Integrals depending on a parameter

2.57 $

Problem: Investigate the uniform convergence of the integral on the set \( Y \). \[ \int_{0}^{\infty} e^{-\alpha x} \tan ^{-1}\left(\alpha x^{2}\right) d x, \quad Y=[0 ;+\infty) . \]

9.2.29 Integrals depending on a parameter

2.57 $

Problem: Investigate the uniform convergence of the integral on the set \( Y \). \[ \int_{0}^{1} \frac{x^{2} \alpha-\alpha^{3}}{\sqrt{x}\left(\alpha^{2}+x^{2}\right)^{2}} d x, \quad Y=(0 ; 1) . \]

9.2.30 Integrals depending on a parameter

2.57 $

Problem: Prove that all elements of order 11 are conjugate in \( S_{11} \). Note: the element \( b \) of the group \( G \) is conjugate of \( a \) through \( g \), if \( b=g a g^{-1} \).

1.3.11 Permutation group

3.09 $

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