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Magazine Problem 37
Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{1}^{\infty} \frac{(\sin x)^{2}}{x^{2}} d x\)
Problem 37
Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int x \sqrt{4-9 x^{2}} d x\)
Problem 38
Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{2}^{\infty} \frac{1}{x(x+1)} d x\)
Problem 38
Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int_{0}^{3} \frac{d x}{\left(4+x^{2}\right)^{\frac{3}{2}}}\)
Problem 39
Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{2}^{\infty} \frac{2}{x^{2} \ln x} d x\)
Problem 39
Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{\sqrt{x^{2}-4}}{x} d x\)
Problem 40
Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int \frac{x}{\sqrt{x^{2}-1}} d x\)
Problem 40
Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{1}^{\infty} \frac{1}{\sqrt{x^{7}+1}} d x\)
Problem 41
Evaluate the integrals. Not all require a trigonometric substitution. Choose the simplest method of integration. \(\int_{\frac{2}{\sqrt{3}}}^{2} \frac{\sqrt{x^{2}-1}}{x} d x\)
Problem 41
Use the comparison theorem to determine whether the integral is convergent or divergent. \(\int_{0}^{\infty} \sin x e^{-x} d x\)
