What is the integral of $x*sqrt(25+x^2)$ ?

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Answer 1 · 2015-07-29T17:34:59

Notice how $sqrt(25 + x^2) prop sqrt(a^2 + x^2)$ , which implies $x = atantheta$ with $a = 5$ .

If we let:

$x = 5tantheta$
$dx = 5sec^2thetad theta$
$sqrt(25 + x^2) = sqrt(5^2 + 5^2tan^2theta) = 5sectheta$

Then we get:

$= int 5tantheta*5sectheta*5sec^2thetad theta$

$= 125int tanthetasecthetasec^2thetad theta$

Then, if we let:

$u = sectheta$
$du = secthetatanthetad theta$

we get:

$= 125int u^2du$

$= 125(u^3/3)$

$= 125((sec^3theta)/3)$

$= ((5^3sec^3theta)/3)$

$= ((5sectheta)^3)/3$

$= ((sqrt(25 + x^2))^3)/3$

$= color(blue)(((25 + x^2)^("3/2"))/3 + C)$

What is the integral of x*sqrt(25+x^2)x*sqrt(25+x^2)?