For trigonometric substitution to solve the above integral, fill in the blanks below using the picture of the triangle given.
integral (x^3)/(sqrt(25+9x^2)) dx / l / l / l / l / l / l / l / l / l side a / l / l side b *** sorry hard to draw on the computer side a=3x side b=? side c=? 3x/5=? (3/5) dx=? (sqrt(25+9x^2))/(5)=?

Substitute so that .

Then you have



Expand the denominator and simplify.



where the last simplification only holds for certain intervals.

Then the integral is



To compute the remaining integral, use the Pythagorean expansion for :



Now substituting so that allows you to rewrite the integral as



Back-substitute:






Simplify the trigonometric terms. Your reference triangle involves an angle with the side opposite the angle having length and the side adjacent to it having length . This means the missing length is .

The secant ratio for this angle is then .

So the antiderivative is equivalent to