IMathAS-libretexts

Evaluate the integral:

\displaystyle \int\frac{\sqrt{{{4}{x}^{{2}}-{25}}}}{{x}^{{3}}}{\left.{d}{x}\right.}

(A) Which trig substitution is correct for this integral?

$\displaystyle {x}=\frac{{1}}{{4}}{\sec{{\left(\theta\right)}}}$
$\displaystyle {x}=\frac{{25}}{{4}}{\sec{{\left(\theta\right)}}}$
$\displaystyle {x}=\frac{{5}}{{2}}{\sec{{\left(\theta\right)}}}$
$\displaystyle {x}=\frac{{5}}{{2}}{\sin{{\left(\theta\right)}}}$
$\displaystyle {x}=\frac{{2}}{{5}}{\sec{{\left(\theta\right)}}}$

(B) Which integral do you obtain after substituting for

\displaystyle {x}

?
Note: to enter

\displaystyle \theta

, type the word theta.

\displaystyle \int

\displaystyle {d}\theta

\displaystyle \theta

?
+ C

(D) What is the value of the original integral in terms of

\displaystyle {x}

?

Note: WAMAP does not recognize the inverse secant (arcsec) function. You will need to use another inverse trig function when you fill in your answer below.
+ C

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