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Problem: Select the main part of the form \( C(x+1)^{k} \) of the infinitesimal function: \[ \alpha(x)=\sqrt{\pi}-\sqrt{\cos ^{-1} x}, \quad x \rightarrow-1 . \]

2.12.16 Asymptotic analysis

1.78 $

Problem: Select the main part of the form \( C\left(\frac{1}{x}\right)^{k} \) of the infinitesimal function: \[ \alpha(x)=\tan \frac{\pi x^{3}+2 x-1}{3 x^{3}-x+2}-\sqrt{3}, \quad x \rightarrow+\infty . \]

2.12.17 Asymptotic analysis

1.53 $

3) Problem: Calculate the line integral using the polygonal \( L \) : \[ \int_{L}\left(x^{3}+y\right) d x+\left(x+y^{2}\right) d y \] where 1) \( L: A B C, 2) L: A B C A, A(7 ; 7), B(3 ; 7), C(3 ; 5) \).

9.4.7 Curvilinear integrals

1.78 $

Problem: Calculate the line integral: \( L:\left\{\begin{array}{c}x=t^{2} \\ y=t\end{array}, \quad 1 \leq t \leq 2, \quad \int_{L} y x d x+y^{2} d y\right. \).

9.4.8 Curvilinear integrals

1.02 $

(3) Problem: Calculate the line integral over the closed polygonal \( A B C D A \) using Ostrogradsky-Green formula: \[ \begin{array}{l} A(3 ; 2), \quad B(6 ; 2), \quad C(6 ; 4), \quad D(3 ; 4) \\ \oint_{L} \sqrt{x^{2}+y^{2}} d x+y\left[x y+\ln \left(x+\sqrt{x^{2}+y^{2}}\right)\right] d y \end{array} \]

9.4.9 Curvilinear integrals

2.54 $

T Problem: Calculate the arc length of the curve given in Cartesian coordinates: \[ x=\ln \cos y, \quad 0 \leq y \leq \frac{\pi}{3} . \]

9.4.10 Curvilinear integrals

1.02 $

3 Problem: Calculate the arc length of the curve given in polar coordinates: \[ r=3(1+\sin a),-\frac{\pi}{6} \leq a \leq 0 \]

9.4.12 Curvilinear integrals

1.27 $

Problem: Prove that the integral \( I(\alpha) \) converges uniformly on the set \( \mathrm{E} \), if: \[ I(\alpha)=\int_{0}^{+\infty} \cos x^{\alpha} d x, \mathrm{E}=\left[\alpha_{0} ;+\infty\right), \alpha_{0}>1 . \]

9.2.13 Integrals depending on a parameter

2.54 $

Problem: Investigate the following integral for uniform convergence: \[ I(\alpha)=\int_{0}^{+\infty} \frac{\sin e^{x}}{1+x^{\alpha}} d x . \]

9.2.14 Integrals depending on a parameter

2.54 $

Problem: Investigate the following integral for uniform convergence: \[ I(\alpha)=\int_{0}^{+\infty} \sqrt{\alpha} \cdot e^{-\alpha x^{2}} d x . \]

9.2.15 Integrals depending on a parameter

2.54 $

Problem: Investigate the continuity of the function \( F(y) \) on the set \( Y \) : \[ F(y)=\int_{0}^{\frac{\pi}{2}} \frac{d x}{x y-\sin x+1}, \quad Y=(0 ;+\infty) . \]

9.2.16 Integrals depending on a parameter

2.03 $

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

9.2.17 Integrals depending on a parameter

3.31 $

Problem: Using the Dirichlet/Frullani/Fresnel/Euler/Poisson/Laplace integrals, calculate the following integral: \[ I=\int_{0}^{+\infty} \frac{\sin x-x \cos x}{x^{3}} d x . \]

9.2.18 Integrals depending on a parameter

4.32 $

Problem: Using the Euler integrals calculate the integral: \[ I=\int_{0}^{+\infty} \frac{\ln x}{\sqrt[3]{x}(x+2)} d x \]

9.2.19 Integrals depending on a parameter

3.81 $

Problem: Find the convolution of the functions \( f(x) \) and \( g(x) \), if the function \( f(x) \) takes a value, equal to zero, when \( x \notin\left[x_{1} ; x_{4}\right] \), and when \( x \in\left[x_{1} ; x_{4}\right] \) its graph consists of links of the broken-line \( A B C D \) : \[ \begin{array}{l} A\left(x_{1} ; 0\right), \quad B\left(x_{2} ; a\right), \quad C\left(x_{3} ; b\right), \quad D\left(x_{4} ; b\right) . \\ x_{1}=-2, \quad x_{2}=1, \quad x_{3}=3, \quad x_{4}=4, \quad a=2, \quad b=-1 . \end{array} \] The function \( g(x) \) has the form \[ g(x)=\left\{\begin{array}{ll} 0, & x<0 \\ 1, & 0 \leq x<1 \\ 0, & x \geq 0 \end{array}\right. \]

11.3.4 Convolution of functions

7.63 $

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