Question

level 1: basic core content
the questions in level 1 are introductory problems. the hints contain links to videos covering this content.
question 1 (1 point)
which of the following type 1 improper integrals converge? (select all that apply.)
$\int_{2}^{\infty} \frac{1}{x} d x$
$\int_{5}^{\infty} \frac{1}{x^{9}} d x$
$\int_{4}^{\infty} \frac{1}{x^{1 / 3}} d x$
$\int_{1}^{\infty} x^{3} d x$
$\int_{-\infty}^{\infty} x \sin \left(x^{2}\
ight) d x$
view hint for question 1

Explanation:

Step1: Test $\int_{2}^{\infty}\frac{1}{x}dx$

Type 1 integral, $p=1$. Diverges (p-test: $p\leq1$ diverges).

Step2: Test $\int_{5}^{\infty}\frac{1}{x^9}dx$

$p=9>1$. Converges (p-test: $p>1$ converges).

Step3: Test $\int_{4}^{\infty}\frac{1}{x^{1/3}}dx$

$p=1/3\leq1$. Diverges.

Step4: Test $\int_{1}^{\infty}x^3dx$

Antiderivative $\frac{x^4}{4}\to\infty$. Diverges.

Step5: Test $\int_{-\infty}^{\infty}x\sin(x^2)dx$

Substitute $u=x^2$, $du=2xdx\Rightarrow xdx=du/2$. Integral becomes $\frac{1}{2}\int_{-\infty}^{\infty}\sin u du$. $\lim_{a\to-\infty,b\to\infty}\frac{1}{2}[-\cos u]_{a}^{b}=\frac{1}{2}(-\cos b+\cos a)$, which oscillates. Diverges.

Answer:

$\int_{5}^{\infty}\frac{1}{x^9}dx$