91Ó°ÊÓ

Explain why each of the following integrals is improper. (a) \( \displaystyle \int_1^2 \frac{x}{x - 1}\ dx \) (b) \( \displaystyle \int_0^\infty \frac{1}{1 + x^3}\ dx \) (c) \( \displaystyle \int_{-\infty}^\infty x^2 e^{-x^2}\ dx \) (d) \( \displaystyle \int_0^{\frac{\pi}{4}} \cot x\ dx \)

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. \( \displaystyle \int \frac{dx}{x^2 \sqrt{4 - x^2}} \) \( x = 2 \sin \theta \)

Evaluate the integral. \( \displaystyle \int \sin^2 x \cos^3 x dx \)

Evaluate the integral using integration by parts with the indicated choices of \( u \) and \( dv \). \( \displaystyle \int xe^{2x} \) ; \( u = x \) , \( dv = e^{2x} dx \)

Evaluate the integral. \( \displaystyle \int \frac{\cos x}{1 - \sin x}\ dx \)

Write out the form of the partial fraction decomposition of the function (as in Example 7). Do not determine the numerical values of the coefficients. (a) \( \dfrac{4 + x}{(1 + 2x)(3 - x)} \) (b) \( \dfrac{1 - x}{x^3 + x^4} \)

Evaluate the integral. \( \displaystyle \int \sin^3 \theta \cos^4 \theta d \theta \)

Evaluate the integral using the indicated trigonometric substitution. Sketch and label the associated right triangle. \( \int \frac{ x^3 }{ \sqrt{x^2 + 4}\ } dx \) \( x = 2 \tan \theta \)

Evaluate the integral using integration by parts with the indicated choices of \( u \) and \( dv \). \( \displaystyle \int \sqrt{x} \ln x dx \) ; \( u = \displaystyle \ln x \) , \( dv = \sqrt{x} dx \)

Evaluate the integral. \( \displaystyle \int_0^1 (3x + 1)^{\sqrt{2}}\ dx \)