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Problem: Find the angle between the straight line \( (A B) \) and the plane \( (C D E) \), where \( A(1 ; \Gamma ; 2), B(-1 ; H ; 1) \), \( C(\Gamma ; 0 ; 4), D(2 ; \mathrm{H} ; 1), E(\mathrm{H} ; \Gamma ; 3) \), where \( \Gamma=\mathrm{H}=2 \). First write the equations of the straight line and the plane.

3.4.5 Tangents and normals

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Problem: Change the order of integration in iterated integral: \[ \int_{-2}^{-1} d x \int_{-(x+2)}^{0} f d y+\int_{-1}^{0} d y x \int_{\sqrt[3]{x}}^{0} f d y . \]

9.1.1 Double integrals

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Problem: Calculate the values of double integrals: 1) \( \iint_{(D)}\left(8 x y+18 x^{2} y^{2}\right) d x d y, \quad D:\left\{\begin{array}{c}x=1 \\ y=\sqrt[3]{x} \\ y=-x^{2}, \quad(x \geq 0)\end{array}\right. \) 2) \( \iint 4 y^{2} \sin 2 x y d x d y, D:\left\{\begin{array}{c}x=0 \\ y=\sqrt{2 \pi} \\ y=2 x\end{array}\right. \)

9.1.2 Double integrals

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Problem: Calculate the mass of the plate \( D \) with density \( \rho \) : \[ D:\left\{\begin{array}{l} x^{2}+y^{2}=4 \\ x^{2}+y^{2}=16 \end{array}, \quad \rho=\frac{2}{x^{2}+y^{2}}\right. \]

9.1.3 Double integrals

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Problem: Explore the integral for uniform convergence with respect to the parameter in the region \( E \) : \[ \begin{array}{l} \int_{0}^{\infty} \sin (\alpha \sinh x) d x, \\ E=\left[\frac{1}{2}, \infty\right), \quad \sinh x=\frac{e^{x}-e^{-x}}{2} . \end{array} \]

9.2.1 Integrals depending on a parameter

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Problem: Define the region of existence and express the integral through the Euler integrals: \[ I=\int_{0}^{\pi} \frac{\sin ^{\alpha-1} x}{(1+\beta \cos x)^{\alpha}} d x, \quad 0<|\beta|<1 . \]

9.2.2 Integrals depending on a parameter

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Problem: Prove the formula: \[ I=\int_{0}^{+\infty} x e^{-x^{3}} d x \int_{0}^{+\infty} e^{-x^{3}} d x=\frac{2 \pi \sqrt{3}}{27} \]

9.2.3 Integrals depending on a parameter

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3) Problem: Calculate the integral: \[ \int e^{5 x^{2}+3 x+1} d x \]

9.3.1.1 Indefinite integrals

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8) Problem: Examine the improper integral for convergence: \[ \int_{0}^{+\infty} \sin x d x \]

9.3.2.1 Improper integrals

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3) Problem: Investigate the convergence of an improper integral of an unbounded function: \[ \int_{0}^{1} \frac{d x}{1-\sqrt[3]{x}} \]

9.3.2.2 Improper integrals

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al Problem: Examine the improper integral of an unbounded function for convergence: \[ I=\int_{0}^{\frac{1}{8}} \frac{\sin ^{-1}\left(x^{2}+x^{5}\right)}{x \ln ^{2}(1-x)} d x \]

9.3.2.3 Improper integrals

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3 Problem: Calculate the integral: \[ \int_{-2 \pi}^{2 \pi} \sin ^{3} 2 x \sqrt{100-x^{2}} d x \]

9.3.3.1 Definite Integrals

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(a) Problem: Calculate the integral: \[ \int_{0}^{1} \frac{\sqrt{x} d x}{4-x} \]

9.3.3.2 Definite Integrals

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Problem: Calculate the integral: 1) \( \int_{-3}^{3} \frac{x^{2} \sin 2 x}{x^{2}+1} d x \) 2) \( \int_{0}^{2} \frac{x^{5} d x}{\sqrt{4-x^{2}}} \).

9.3.3.3 Definite Integrals

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Problem: Calculate the integral: 1) \( \int \frac{d x}{(x-1)^{2}(x-2)} \), 2) \( \int \cos \frac{x}{2} \cos \frac{x}{3} d x \), 3) \( \int \frac{\sqrt[6]{x}+1}{\sqrt[6]{x^{7}}+\sqrt[4]{x^{5}}} d x \).

9.3.1.2 Indefinite integrals

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