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Integration by parts?
I am having difficulties integrating by parts. I have been trying these two problems over and over with no results. If you could help me with either or both problems that would be great.
1) the integral of 5theta^3 cos(theta^2) dx between square root (pi/2) and square root (pi)
2) integral of 5(ln(7x))^2 dx
1 Answer
- Old TeacherLv 79 years agoFavorite Answer
1) did you mean dx , or dtheta? I will use x for all.
INT[ 5x^3 cos(x^2)] dx = (5/2) INT [ x^2 * 2xcos(x^2) ] dx
U= x^2 ; dV= 2xcos(x^2)dx
Du = 2xdx ; V= sin(x^2)
(5/2) [ x^2 sin(x^2) - INT [ 2x sin(x^2)dx]
= (5/2)[ x^2 sin(x^2) + cos(x^2)] | [sqr(pi/2), sqr(pi)]
= (5/2) [ pi*sin(pi) + cos(pi) - (pi/2)sin(pi/2) - cos(pi/2)]
= (5/2)[ -1- pi/2]
------
2) if you have already derived the formula for integrating ln(x)= x* ln(x)-x
Then let U = (ln 7x)^2 ; dv= dx
Du = 2(ln7x) * (1/x) dx and v= x
5[ INT (ln 7x)^2 dx] = 5[ x* (ln 7x)^2 - INT ( 2 ln(7x)] dx
= 5[ x(ln 7x)^2 -(2/7)[ (7x) ln(7x) -(7x)] ]
Which you can simplify.
---- if you do not have this formula, then integrate by parts again. Where u = ln(7x)
Hoping this helps!
