# How integrals are convergent? int_(0)^(oo)sin17x dx

Question
Applications of integrals
How integrals are convergent?
$$\displaystyle{\int_{{{0}}}^{{\infty}}}{\sin{{17}}}{x}{\left.{d}{x}\right.}$$

2020-11-13
Step 1
Since there are multiple questions, so we will be answering only the first one.
To deal with such type of Improper Integrals, we will replace the infinity with a variable (usually t), do the integral and then take the limit of the result as t goes to infinity.
We will call these integrals convergent if the associated limit exists and is a finite number (i.e. it’s not plus or minus infinity) and divergent if the associated limit either doesn’t exist or is (plus or minus) infinity.
$$\displaystyle{\int_{{{0}}}^{{\infty}}}{\sin{{17}}}{x}{\left.{d}{x}\right.}=\lim_{{{t}\rightarrow\infty}}{\int_{{0}}^{{t}}}{\sin{{17}}}{x}{\left.{d}{x}\right.}$$
$$\displaystyle\lim_{{{t}\rightarrow\infty}}{\int_{{0}}^{{t}}}{\sin{{17}}}{x}{\left.{d}{x}\right.}=\lim_{{{t}\rightarrow\infty}}{\left[\frac{{1}}{{17}}{\left(-{\cos{{\left({17}{t}\right)}}}+{1}\right)}\right]}$$
=limit does not exist
Thus, $$\displaystyle{\int_{{0}}^{{\infty}}}{\sin{{17}}}{x}{\left.{d}{x}\right.}$$ is not convergent.

### Relevant Questions

Show that the differential forms in the integrals are exact. Then evaluate the integrals.
$$\displaystyle{\int_{{{\left({1},{1},{2}\right)}}}^{{{\left({3},{5},{0}\right)}}}}{y}{z}{\left.{d}{x}\right.}+{x}{z}{\left.{d}{y}\right.}+{x}{y}{\left.{d}{z}\right.}$$
Which of the following integrals are improper integrals?
1.$$\displaystyle{\int_{{{0}}}^{{{3}}}}{\left({3}-{x}\right)}^{{{2}}}{\left\lbrace{3}\right\rbrace}{\left.{d}{x}\right.}$$
2.$$\displaystyle{\int_{{{1}}}^{{{16}}}}{\frac{{{e}^{{\sqrt{{{x}}}}}}}{{\sqrt{{{x}}}}}}{\left.{d}{x}\right.}$$
3.$$\displaystyle{\int_{{{1}}}^{{\propto}}}{\frac{{{3}}}{{\sqrt{{{3}}}{\left\lbrace{x}\right\rbrace}}}}{\left.{d}{x}\right.}$$
4.$$\displaystyle{\int_{{-{2}}}^{{{2}}}}{3}{\left({x}+{1}\right)}^{{-{1}}}{\left.{d}{x}\right.}$$
a) 1 only
b)1 and 2
c)3 only
d)2 and 3
e)1,3 and 4
f)All of the integrals are improper
Evaluate the following definite integrals
$$\displaystyle{\int_{{{0}}}^{{{1}}}}{\left({x}^{{{4}}}+{7}{e}^{{{x}}}-{3}\right)}{\left.{d}{x}\right.}$$
Evaluate the following definite integrals
$$\displaystyle{\int_{{{0}}}^{{{1}}}}{x}{e}^{{{\left(-{x}^{{{2}}}+{2}\right)}}}{\left.{d}{x}\right.}$$
Show that the differential forms in the integrals are exact. Then evaluate the integrals.
$$\displaystyle{\int_{{{1},{1},{2}}}^{{{3},{5},{0}}}}{y}{z}{\left.{d}{x}\right.}+{x}{z}{\left.{d}{y}\right.}+{x}{y}{\left.{d}{z}\right.}$$
Evaluate the following integrals.
$$\displaystyle{\int_{{-{2}}}^{{-{1}}}}\sqrt{{-{4}{x}-{x}^{{{2}}}}}{\left.{d}{x}\right.}$$
$$\displaystyle{\int_{{-{2}}}^{{{2}}}}{\left({3}{x}^{{{4}}}-{2}{x}+{1}\right)}{\left.{d}{x}\right.}$$
$$\displaystyle{\int_{{-{{2}}}}^{{-{{1}}}}}\sqrt{{-{4}{x}-{x}^{{2}}}}{\left.{d}{x}\right.}$$
$$\displaystyle{\int_{{-{{2}}}}^{{2}}}{\left({3}{x}^{{4}}-{2}{x}+{1}\right)}{\left.{d}{x}\right.}$$
$$\displaystyle{\int_{{{0}}}^{{{1}}}}{\frac{{{4}{r}{d}{r}}}{{\sqrt{{{1}-{r}^{{{4}}}}}}}}$$