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

9.3.1.3 Indefinite integrals

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(3) Problem: Find the arc length of the curve which is specified parametrically: \[ L: r=2 \cos ^{3} \frac{\varphi}{3}, \varphi \in\left[0 ; \frac{3 \pi}{2}\right] \]

9.4.1 Curvilinear integrals

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(3) Problem: Find the mass of the curve, given parametrically: \( \ell: y=\ln \cos x, x \in\left[0 ; \frac{\pi}{3}\right] \), density \( \rho=e^{y} \).

9.4.2 Curvilinear integrals

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(a) Problem: Calculate the line integral from point \( A \) to point \( C \) along three different paths: \[ \begin{array}{l} A(0 ; 1), \quad B(3 ; 1), \quad C(3 ; 10) \\ I=\int_{\ell}(2 y+x) d x+(2 x-1) d y \end{array} \]

9.4.3 Curvilinear integrals

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(a) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{cc} x=19 \sqrt{2 y}, & x=4 \sqrt{2 y} \\ z=0, & z+y=2 \end{array} .\right. \]

9.5.1 Volume of a solid

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3) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{c} x^{2}+y^{2}=3 y \\ x^{2}+y^{2}=6 y \\ z=0, \quad z=\sqrt{x^{2}+y^{2}} \end{array}\right. \]

9.5.2 Volume of a solid

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(3) Problem: Calculate the volume of the solid \( V \), bounded by the given surfaces: \[ V:\left\{\begin{array}{c} x=y^{2}-2, \quad x=-4 y^{2}+3 \\ z=\sqrt{16-x^{2}-y^{2}}+2, \quad z=\sqrt{16-x^{2}-y^{2}}-1 \end{array}\right. \]

9.5.3 Volume of a solid

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Problem: For a curve given parametrically, find: 1. Vectors of the accompanying trihedron at the point \( t=t_{0} \); 2. Planes and lines of the accompanying trihedron at the point \( t=t_{0} \); 3. Tangent lines parallel to coordinate planes; 4. Contiguous planes perpendicular to the coordinate axes; 5. Curvature and torsion of a curve at the point \( t=t_{0} \)

7.12 Differential geometry

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Problem: Plot the graph of the function \( y=\left|\left(\frac{1}{2}\right)^{|x+1|}-3\right| \) using elementary transformations of the graph of the function \( y=\left(\frac{1}{2}\right)^{x} \).

2.5.1 Plotting functions

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Problem: Examine the function by methods of differential calculus and plot the graph: \[ f(x)=\frac{x^{3}}{x^{2}+1} \text {. } \]

2.4.2 Graphing functions using derivatives

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Problem: Examine the function by methods of differential calculus and plot the graph: \[ f(x)=x^{2} e^{x} \text {. } \]

2.4.3 Graphing functions using derivatives

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Problem: Examine the function by methods of differential calculus and plot the graph: \[ f(x)=x+\ln x \text {. } \]

2.4.4 Graphing functions using derivatives

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Problem: Examine the function by methods of differential calculus and plot the graph: \[ y=\frac{-2 x^{3}-x^{2}+2 x+3}{x^{2}-2 x-3} . \]

2.4.5 Graphing functions using derivatives

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Problem: Examine the function, given by the parametric equations, in accordance with the given scheme. Plot its graph. \[ \begin{array}{l} f(t)=t^{3}-3 t, \quad g(t)=-3 t^{3}-4 t^{2}+t \\ \left\{\begin{array}{l} x=f(t) \\ y=g(t) \end{array}-\infty

2.4.6 Graphing functions using derivatives

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Problem: Investigate the function by methods of differential calculus and plot the graph: \[ y=x e^{-x^{2}} \text {. } \]

2.4.7 Graphing functions using derivatives

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