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Problem: Calculate the integral using residues: \[ \int_{-\infty}^{+\infty} \frac{\cos x d x}{\left(x^{2}-4 x+5\right)^{2}} \text {. } \]

10.7.8 Calculating integrals of a real variable using residues

3.06 $

Problem: Applying the Cauchy residue theorem, calculate the integral. \[ \int_{0}^{\pi} \frac{\sin ^{4} \varphi}{1+\cos ^{2} \varphi} d \varphi \text {. } \]

10.7.9 Calculating integrals of a real variable using residues

3.06 $

Problem: Applying the Cauchy residue theorem, calculate the integral. \[ \int_{0}^{2 \pi} \frac{\cos ^{2} \varphi}{13+12 \cos \varphi} d \varphi . \]

10.7.10 Calculating integrals of a real variable using residues

3.06 $

Problem: Applying the Cauchy residue theorem, calculate the integral. \[ \int_{0}^{\pi} \frac{\cos ^{4} \varphi}{1+\sin ^{2} \varphi} d \varphi \text {. } \]

10.7.11 Calculating integrals of a real variable using residues

3.83 $

Problem: Applying the Cauchy residue theorem, calculate the integral. \[ \int_{0}^{2 \pi} \frac{\sin ^{2} \varphi}{5+4 \sin \varphi} d \varphi \text {. } \]

10.7.12 Calculating integrals of a real variable using residues

3.06 $

Problem: Appropriately applying the Cauchy residue theorem, find the principal value of the integral: \[ \int_{-\infty}^{\infty} \frac{x \cos x}{x^{2}-5 x+6} d x \text {. } \]

10.7.13 Calculating integrals of a real variable using residues

5.11 $

Problem: Appropriately applying the Cauchy residue theorem, find the principal value of the integral: \[ \int_{-\infty}^{\infty} \frac{\sin x}{\left(x^{2}+4\right)(x-1)} d x . \]

10.7.14 Calculating integrals of a real variable using residues

5.11 $

Problem: Appropriately applying the Cauchy residue theorem, find the principal value of the integral: \[ \int_{-\infty}^{+\infty} \frac{\cos \lambda x}{1+x^{3}} d x, \lambda \in \mathbb{R} \text {. } \]

10.7.15 Calculating integrals of a real variable using residues

7.66 $

Problem: Appropriately applying the Cauchy residue theorem, find the principal value of the integral: \[ \int_{-\infty}^{+\infty} \frac{\cos \lambda x}{1-x^{4}} d x, \lambda \in \mathbb{R} \]

10.7.16 Calculating integrals of a real variable using residues

6.38 $

Problem: Calculate the integral using residues: \[ \int_{0}^{2 \pi} \frac{d t}{7+\sqrt{13} \sin t} \text {. } \]

10.7.17 Calculating integrals of a real variable using residues

0 $

Problem: Calculate the integral using residues: \[ \int_{0}^{2 \pi} \frac{d t}{(5+2 \sqrt{6} \cos t)^{2}} \text {. } \]

10.7.18 Calculating integrals of a real variable using residues

3.06 $

Problem: Calculate the integral and characterize all singular points of the functions under the integral sign (including the point \( \infty \) ): \[ \int_{\partial \Omega} \frac{z^{2} \sin ^{2} \frac{1}{z}}{(z-1)(z-2)} d z \text {, where } \Omega=\{|z|<2\} . \]

10.1.15 Integral of a complex variable

3.06 $

Problem: Calculate the integral and characterize all singular points of the functions under the integral sign (including point \( \infty \) ). \[ \int_{\partial \Omega} \frac{z^{3}}{z+1} e^{\frac{1}{z}} d z \text {, where } \Omega=\{|z|<2\} \text {. } \]

10.1.16 Integral of a complex variable

3.06 $

Problem: Calculate the integral and characterize all singular points of the functions under the integral sign (including point \( \infty \) ). \[ \int_{\partial \Omega} z \sin \frac{z+1}{z-1} d z \text {, where } \Omega=\{|z|<2\} . \]

10.1.17 Integral of a complex variable

3.06 $

Problem: Calculate the integral and characterize all singular points of the functions under the integral sign (including point \( \infty \) ). \[ \int_{\partial \Omega} \frac{\sin z}{\left(z^{3}-z\right)(z-i)} d z \text {, where } \Omega=\{|z-1|<1\} \text {. } \]

10.1.18 Integral of a complex variable

3.06 $

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