Answer to Question #139778 in Calculus for Moel Tariburu

Question #139778

Evaluate the integral ∫x^3 sin⁡〖x〗dx


1
Expert's answer
2020-10-25T18:42:58-0400

x3sinxdx\int x^3sinx\, dx



Using the formula,Udv=UVVduUsing\ the\ formula, \int Udv = UV -\int Vdu


U=x3;dv=sinxdxdudx=3x2;dv=sinxdxdu=3x2dx;V=cosx\begin{aligned} U = x^3 \qquad &; \qquad dv=sinx\,dx \\ \frac{du}{dx}=3x^2\qquad &; \qquad \int dv = \int sinx\, dx\\ du= 3x^2\,dx\qquad &; \qquad V = -cosx \end{aligned}


x3sinxdx=(x3)(cosx)(cosx)(3x2dx)\therefore \int x^3sinx\, dx = (x^3)(-cosx) - \int(-cosx)(3x^2\,dx)


x3sinxdx=x3cosx(3x2cosxdx)\int x^3sinx\, dx = -x^3\,cosx - \int( -3x^2cosx\,dx)


x3sinxdx=x3cosx(3)x2cosxdx\int x^3sinx\, dx = -x^3\,cosx - (-3)\int x^2cosx\,dx


x3sinxdx=x3cosx+3x2cosxdx (i)\int x^3sinx\, dx = -x^3\,cosx +3 \int x^2cosx\,dx\ --(i)




x2cosxdx=\int x^2cosx\,dx =


U=x2;dv=cosxdxdudx=2x;dv=cosxdxdu=2xdx;V=sinx\begin{aligned} U = x^2\qquad &; \qquad dv = cosx\,dx\\ \frac{du}{dx}= 2x\qquad &; \qquad \int dv = \int cosx\,dx\\ du = 2x\,dx\qquad &; \qquad V = sinx \end{aligned}


x2cosxdx=(x2)(sinx)(sinx)(2xdx)\therefore \int x^2cosx\, dx = (x^2)(sinx) - \int(sinx)(2x\,dx)


x2cosxdx=x2sinx2xsinxdx (ii)\int x^2cosx\, dx = x^2\,sinx - 2\int xsinx\,dx\ ---(ii)




xsinxdx=\int xsinx\,dx =


U=x;dv=sinxdxdudx=1;dv=sinxdxdu=dx;V=cosx\begin{aligned} U = x\qquad &; \qquad dv = sinx\,dx\\ \frac{du}{dx}= 1\qquad &; \qquad \int dv = \int sinx\,dx\\ du = dx\qquad &; \qquad V = -cosx \end{aligned}


xsinxdx=(x)(cosx)(cosx)(dx)\therefore \int xsinx\, dx = (x)(-cosx) - \int(-cosx)(dx)


xsinxdx=xcosx+cosxdx\int xsinx\, dx = -xcosx + \int cosx\,dx


xsinxdx=xcosx+sinx\int xsinx\, dx = -xcosx + sinx


Substituting the value of xsinxdx\int xsinx\,dx into equation (ii),


x2cosxdx=x2sinx2xsinxdx\int x^2cosx\, dx = x^2\,sinx - 2\int xsinx\,dx


x2cosxdx=x2sinx2(xcosx+sinx)\int x^2cosx\, dx = x^2\,sinx - 2(-xcosx+sinx)


x2cosxdx=x2sinx+2xcosx2sinx\int x^2cosx\, dx = x^2\,sinx +2xcosx - 2sinx



Substituting the value of x2cosxdx\int x^2cosx\,dx into equation (i), we have


x3sinxdx=x3cosx+3x2cosxdx\int x^3sinx\, dx = -x^3\,cosx +3 \int x^2cosx\,dx


x3sinxdx=x3cosx+3(x2sinx+2xcosx2sinx)\int x^3sinx\, dx = -x^3\,cosx +3 (x^2\,sinx +2xcosx - 2sinx)


x3sinxdx=x3cosx+3x2sinx+6xcosx6sinx+C\int x^3sinx\, dx = -x^3\,cosx +3x^2\,sinx +6xcosx - 6sinx + C


The integral of x3sinx=x3cosx+3x2sinx+6xcosx6sinx+C\therefore The\ integral\ of\ x^3sinx = -x^3\,cosx +3x^2\,sinx +6xcosx - 6sinx + C





Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!

Leave a comment