Best solutions to - "Improper integrals Evaluate the following integrals or stat..."

UkusakazaL

Answered question

2021-11-04

Improper integrals Evaluate the following integrals or stat that they diverge.

\int_{1}^{\infty} \frac{3 x^{2} + 1}{x^{3} + x} d x

insonsipthinye

Skilled2021-11-05Added 83 answers

Step 1

\int_{1}^{\infty} \frac{3 x^{2} + 1}{x^{3} + x} d x

Substitute the value of

(x^{3} + x) = t

and differentiate it w.r.t x

\frac{d}{d x} (x^{3} + x) = \frac{d t}{d x}

(3 x^{2} + 1) d x = d t

Lower limit at x=1 is t=2.
Upper limit at

x = \infty i s t = \infty

Step 2
Therefore, given integral becomes

\int_{2}^{\infty} \frac{1}{t} d t

\Rightarrow {[\ln | t |]}_{2}^{\infty}

\Rightarrow \ln | \infty | - \ln | 2 |

\Rightarrow \infty

Therefore, the given integral is diverge.

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