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Question: How do you find the integral of \(\left[ {{x^4}\sin x\,dx} \right]\)?...

How do you find the integral of [x4sin⁡x dx]\left[ {{x^4}\sin x\,dx} \right]?

Explanation

Solution

In this question, we have to do by-parts integration. But here we have to do the by-parts integration not once but several times till we get to the final answer. So, for that we should know the formula of by-parts integration and how we apply it.

Formula used: ∫u v dx=u∫v dx−∫(dudx∫vdx)dx\int {u\,v\,dx = u\int {v\,dx - \int {\left( {\frac{{du}}{{dx}}\int {vdx} } \right)dx} } }

Complete step by step answer:
In the above question, it is given that we have to find the integral of [x4sin⁡x dx]\left[ {{x^4}\sin x\,dx} \right].
In this question, we need to integrate by parts repeatedly, to reduce the degree of x.
∫x4sin⁡x dx=x4∫sin⁡xdx−∫[ddxx4∫sin⁡xdx]\int {{x^4}\sin x\,dx = } {x^4}\int {\sin xdx - \int {\left[ {\frac{d}{{dx}}{x^4}\int {\sin xdx} } \right]} }
⇒x4(−cos⁡x)−[∫4x3(−cos⁡x)dx]\Rightarrow {x^4}\left( { - \cos x} \right) - \left[ {\int {4{x^3}\left( { - \cos x} \right)dx} } \right]
Now, on simplification we get
⇒−x4cos⁡x+[∫4x3(cos⁡x)dx]\Rightarrow - {x^4}\cos x + \left[ {\int {4{x^3}\left( {\cos x} \right)dx} } \right]
⇒−x4cos⁡x+4x3∫cos⁡dx−[∫ddx(4x3)∫cos⁡xdx]\Rightarrow - {x^4}\cos x + 4{x^3}\int {\cos dx - \left[ {\int {\frac{d}{{dx}}\left( {4{x^3}} \right)\int {\cos xdx} } } \right]}
⇒−x4cos⁡x+4x3sin⁡x−∫12x2sin⁡xdx\Rightarrow - {x^4}\cos x + 4{x^3}\sin x - \int {12{x^2}\sin xdx}
⇒−x4cos⁡x+4x3sin⁡x+12x2cos⁡x−∫24xcos⁡xdx\Rightarrow - {x^4}\cos x + 4{x^3}\sin x + 12{x^2}\cos x - \int {24x\cos xdx}
⇒−x4cos⁡x+4x3sin⁡x+12x2cos⁡x−24xsin⁡x+24∫sin⁡xdx\Rightarrow - {x^4}\cos x + 4{x^3}\sin x + 12{x^2}\cos x - 24x\sin x + 24\int {\sin xdx}
⇒−x4cos⁡x+4x3sin⁡x+12x2cos⁡x−24xsin⁡x+24(−cos⁡x)+C\Rightarrow - {x^4}\cos x + 4{x^3}\sin x + 12{x^2}\cos x - 24x\sin x + 24\left( { - \cos x} \right) + C
⇒−x4cos⁡x+4x3sin⁡x+12x2cos⁡x−24xsin⁡x−24cos⁡x+C\Rightarrow - {x^4}\cos x + 4{x^3}\sin x + 12{x^2}\cos x - 24x\sin x - 24\cos x + C
Therefore, the value of the required integral is ⇒−x4cos⁡x+4x3sin⁡x+12x2cos⁡x−24xsin⁡x−24cos⁡x+C \Rightarrow - {x^4}\cos x + 4{x^3}\sin x + 12{x^2}\cos x - 24x\sin x - 24\cos x + C.

Note: Integration by parts is a special technique of integration of two functions when they are multiplied. This method is also termed as partial integration. Another method to integrate a given function is integration by substitution method. These methods are used to make complicated integrations easy.