How to integrate $\int x\sin {(\sqrt{x})}\, dx$

Solution 1:

Yes, indeed, continue as you did in the comments, treating $\int 6t\sin t \,dt\;$ as a separate integral, use integration by parts, and add (or subtract, if appropriate) that result to your earlier work, and you will end with an expression with no integrals remaining!:

$$\int t^2 \cdot \sin({t})\cdot 2t dt = $$

$$= 2[-\cos(\sqrt x) \cdot x(\sqrt x) + \sin(\sqrt x)\cdot 3x -(\cos(\sqrt x)\cdot6\sqrt x+\sin(\sqrt x)\cdot \sqrt x + \cos (\sqrt x))] + C$$

after substituting $\sqrt x$ for $t$, though I'd suggest finding a way to simplify (combining like terms, etc.)

Solution 2:

Just continue your path of partial integration with the last integral ? The last integral is purely a cosine which is integrable and yields your sollution.