Integrals depending on a parameter

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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 $

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

3.25 $

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

4.25 $

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

3.75 $

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

6.24 $

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

4.99 $

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

4.99 $

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

9.2.23 Integrals depending on a parameter

3 $

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 $

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

4.99 $

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

3.75 $

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

2.5 $

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

2.5 $

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

2.5 $

Problem: Investigate the integral over the parameter for uniform convergence: \[ \int_{0}^{1} \sin \left(x^{a}(\ln x)^{2}\right) d x, \quad a \leq a_{0}<1 . \]

9.2.27 Integrals depending on a parameter

1.5 $

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