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Question Number 86132 by M±th+et£s last updated on 27/Mar/20

$$\int x^{3}sin{(2x^2+6)}^{5}dx$$

Answered by Kunal12588 last updated on 27/Mar/20

$$I=\int x^{3}sin{(2x^2+6)}^{5}dx$$$$let2x^2+6=t$$$$\Rightarrow 4xdx=dt$$$$I=\frac{1}{4}\int (\frac{t}{2}-3)sin{t}^{5}dt$$$$=\frac{1}{4}[(\frac{t}{2}-3)\int sin{t}^5-\frac{1}{2}\int (\int sin{t}^{5}dt)dt]$$$$let\int sin{t}^{5}dt=F\left(t\right)+c$$$$I=\frac{1}{8}(t-6)F\left(t\right)+\frac{1}{8}(ta-6a)-\frac{1}{8}\int (F\left(t\right)+b)dt$$$$=\frac{1}{8}(t-6)F\left(t\right)+\frac{a}{8}t-\frac{b}{8}t-\frac{1}{8}\int F\left(t\right)dt-\frac{3}{4}a$$$$=\frac{1}{8}(t-6)F\left(t\right)+ct-\frac{1}{8}\int F\left(t\right)dt+C$$$$Andnowit‘‘may{be}^{″}easyforthepersons$$$$whoknows\int sin{x}^{5}dx:)$$

Commented by M±th+et£s last updated on 27/Mar/20

$$thankyousir$$

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