How do you integrate x*sin(x)^2?

Answer 1

integrate x*sin(x)^2

let x^2= t => 2x dx= dt => xdx= 1/2 dt

intigrate x*sin(x)^2= intigrate 1/2 sint dt=1/2 intigrate sint dt

=> 1/2 (-cost)+C= -1/2 cost+C = -1/2 cos(x)^2+C

there fore integrate x*sin(x)^2= -1/2 cos(x)^2+C

Answer 2

let u = x^2 , du = 2xdx, => dx = 1/(2x)du

the equation become Integral[1/2sinu]

the integral is -1/2 coxu = -1/2cos(x^2) + C

Answer 3

Apply Integral(u dv)=uv - Integral (v du) formula. Take u as x and dv as sin(x)^2dx.

Answer 4

use bernouli's formula :
integral udv=u v1-u' v2+ u'' v3- u''' v4+..................(until the term becomes 0)
u'-differentiate u once,u''-differentiatye u twice ,so on
v1-integrating v once,v2-integrating v twice,so on
to choose u &dv ,use ILATE formula(I-inverse,L-logrithmic,A-algebraic,T-trigonometric,E-exponential), to choose u in corresponding above order,in ur sum
x->algebraic, sin(x) square-trigonometric expression, so choose
x as u and sin(x) squre as dv
sin(x) square=(1-cos2a)/2

Answer 5

use Integrator of Mathematica

Answer 6

integration by parts (u and v)

Answer 7

cos(2x) = (cosx)^2 - (sinx)^2
or, cos(2x) = 1 - 2*(sinx)^2
or, (sinx)^2 = 0.5* (1 - cos(2x))

replace (sinx)^2 to get the integration of
0.5 * x * (1 - cos(2x))
= 0.5*x - 0.5*x*cos(2x)

first part can be integrated as 0.25 x^2

second part can be solved in UV method