please explain each step clearly (working and explanation) thank you!!

Integration by Parts A Theorem Let p(x) be an abstract function defined on the interval [a, b] with the following properties: Property 1: p(a) = p(b) = 0 Property 2: p"(x) exists for each x in [a,b]. a. Use these properties to prove the equation Ja J. D(x)D"(2) dx = - ('(x) dx . (Hint, integrate the left side by parts and simplify). As another side note, it will be useful to use the integration by parts formula if the integrals have limits: f()8'(x)dx+ /S'(x)g(x) dx = f(x)g(x)la The right side of the equation is evaluated from a to b. Additional Condition b. In addition to the properties above, suppose p" is proportional to p. In symbols. Property 3: p"(x) = Kp(x), for some constant K. Plug this information into the equation you proved above and make a mathematical argument that K must be a negative number.B. Application to Trig Integrals 1dx a. Compute Ja sin x dx cos' x dx 1dx b. Use a pythagorean identity that relates Ja and Ja to Ja . Give a graphical argument to illustrate this. c. In part A above, Property 2 required that p(a) = p(b) = 0. If p(x) = sin(x), we can relax this condition a little bit. Find general conditions on the values of a and b so that p(x) = sin(x) satisfies the equation p(x)p"(x) dx = - (p'(x) dx b - a sin x dx = d. Use your result from parts b. and c. to show that Ja 2 for a qualifying interval [a,b]