# Math engineering integrals

1. Evaluate $\int (\sqrt{(1-x^2)}/x^2)$

2. Evaluate $\int (x^2/(1-x^2)^3/2)$

3. Find the length of the curve $y =4 + ln(sec(x))$, x between 0 and $\pi/4$

1. $\int (\sqrt{(1-x^2})/x^2)dx =$

$x =sin u$, $dx =cos u * du$

$=\int (\sqrt{(1-sin^2(u))}/sin^2(u) *cos(u)*du = \int (cos^2(u)/sin^2(u)*du)=$

$=\int (cotan^2(u)*du) =\int (csc^2(u)-1)du = -cotan(u)-u +C =$

$u = arcsin(x)$

$=-cotan(arcsin(x))-arcsin(x)+ C$

2. $\int (x^2/(1-x^2)^{3/2} *dx) =$

$x=sin(u)$, $dx =cos(u)*du$

$=\int(sin^2(u)/(1-sin^2(u)) *cos(u)du =\int (sin^2(u)*cos(u)/cos^3(u)) du =$

$=\int (sec^2(u))du = tan(u) + C=$

$u = arcsin(x)$

$=tan(arcsin(x)) + C$

3. $y =4+ln(sec(x))$, $arc(ab)$ between $a = 0$ and $b =\pi/4$

length of $arc(ab)$ is $S = \int_a^b \sqrt{(1+ (dy/dx)^2)}*dx$

$dy/dx = tan(x)$

$S = \int_0^{\pi/4} \sqrt{(1+tan^2(x))}dx = \int_0^{\pi/4} (\sqrt{(sec^2(x))} dx=$

$=\int_0^{\pi/4} (sec(x)dx) = log(tan(x) +sec(x))|_0^{\pi/4} =$

$= log (tan(pi/4)+sec(pi/4))/(tan(0)+sec(0)) = log (1 +1.414)/(0 +1) =$

$=log (2.414/1) =0.382$