Posted by W on Monday, January 2, 2012 at 8:15pm.
when all else fails
go to wolfram alpha (Just type that in your browser)
in the box type:
Integrate (1/2)sin(x^(1/2))
after you see the answer, click on "show steps" in the box on the upper right.
integral (sin(sqrt(x)))\/2 dx = sin(sqrt(x))-sqrt(x) cos(sqrt(x))+constant
Possible intermediate steps:\n integral (sin(sqrt(x)))\/2 dx\nFactor out constants:\n = 1\/2 integral sin(sqrt(x)) dx\nFor the integrand sin(sqrt(x)), substitute u = sqrt(x) and du = 1\/(2 sqrt(x)) dx:\n = integral u sin(u) du\nFor the integrand u sin(u), integrate by parts, integral f dg = f g- integral g df, where \n f = u, dg = sin(u) du,\n df = du, g = -cos(u):\n = integral cos(u) du-u cos(u)\nThe integral of cos(u) is sin(u):\n = sin(u)-u cos(u)+constant\nSubstitute back for u = sqrt(x):\n = sin(sqrt(x))-sqrt(x) cos(sqrt(x))+constant
Obviously it does not copy and paste very well so go to the original. Your substitution approach was in fact what was used.
let u= sqrtx
du= 1/2*1/sqrtx dx
dx= 2u du
INT 1/2 sin (sqrtx) dx
INT 1/2 sin u 2u du
int u sin u du=-u cos u + sin u
then change it back to x
-sqrtx cosSqrtx+sinsqrtx
check: take the derivative...
-1cossqrtx/2sqrx+ -sqrtx(-sinsqrtx)1/2sqrx +cossqurx/2sqrtx
-1/2(-sinSqrtx)
1/2 sinSqrtx so it checks.
very cool. Thanks, both of you! Does wolfram alpha give steps for all integration problems?
Pretty much so
your other one would be
integrate 1/(x ln x)
awesome! Thanks Damon ;D