MHB Differentiation under integral sign

WMDhamnekar
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Hello,

How to find formulas for these$\displaystyle\int x^n\sin(x)\, dx, \displaystyle\int x^n\cos(x)\, dx,$ indefinite integrals when $n=1,2,3,4$ using differentiation under the integral sign starting with the formulas

$$\displaystyle\int \cos(tx)\,dx = \frac{\sin(tx)}{t}, \displaystyle\int \sin(tx)\,dx= -\frac{\cos(tx)}{t}$$ for $t > 0.$

I don't have any idea to solve these indefinite integrals except to solve them recursively using integration by parts.
 
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You are required to use "differentiation" under the integral? These can be done directly using "integration by parts".
 
See what happens when you differentiate both sides of those relations with respect to $t$ and then set $t=1$.
 
MountEvariste said:
See what happens when you differentiate both sides of those relations with respect to $t$ and then set $t=1$.

Hello,
I solved the two indefinate integrals using differentiation under integral sign as well as using integration by parts. Under these both method i got the correct answers. For example if we put n=4,the integral to be solved becomes $\displaystyle\int x^4\sin{(tx)}dx =\frac{(4x^3t^3-24tx)*\sin{(tx)}+(-x^4t^4+12x^2t^2-24)*\cos{(tx)}}{t^5}$

Now if we put t=1, the answer will become $(4x^3-24x)*\sin{(x)}+(-x^4+12x^2-24)*\cos{(x)}$

Now how to derive the formula for $\displaystyle\int x^4\sin{(x)}dx?$

In other cases where n=1,2,3, similar problem will persist.
 
Dhamnekar Winod said:
In other cases where n=1,2,3, similar problem will persist.
Perhaps you've figured it out by now, but I don't see any differentiation under integral sign.

$$\displaystyle\int \cos(tx)\,dx = \frac{\sin(tx)}{t}, \displaystyle\int \sin(tx)\,dx= -\frac{\cos(tx)}{t}$$ for $t > 0.$
In these two equalities, differentiate both the RHS and the LHS with respect to $t$. Adjust for constant. Do it again, and again. You'll get all of them.
 
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