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Question Number 33619 by NECx last updated on 20/Apr/18

$$\int x^{5/2}{(1-x)}^{3/2}dx$$

Commented by abdo imad last updated on 20/Apr/18

$$letputI=\int x^{\frac{5}{2}}{(1-x)}^{\frac{3}{2}}dx.ch.x={sin}^{2}tgive$$$$I=\int {sin}^{5}x{cos}^{3}x\left(2sintcost\right)dt$$$$=2\int {cos}^{4}x{sin}^{6}xdx$$$$=2\int {\left(cosxsinx\right)}^4{sin}^{2}xdx$$$$=\frac{1}{8}\int {\left(sin\left(2x\right)\right)}^4\frac{1-cos\left(2x\right)}{2}dx$$$$=\frac{1}{16}\int {\left(\frac{1-cos\left(4x\right)}{2}\right)}^2(1-cos\left(2x\right))dx$$$$=\frac{1}{64}\int (1-2cos\left(4x\right)+{cos}^2\left(4x\right))(1-cos\left(2x\right))dx$$$$=\frac{1}{64}\int (1-2cos\left(4x\right)+\frac{1+cos\left(8x\right)}{2})(1-cos\left(2x\right))dx$$$$=\frac{1}{128}\int (2-4cos\left(4x\right)+1+cos\left(8x\right))(1-cos\left(2x\right))dx$$$$=.....untilwefindthevalue....$$

Commented by abdo imad last updated on 20/Apr/18

$$thismethodisusefulif-1⩽x⩽1.$$

Commented by NECx last updated on 21/Apr/18

$$cantitbecompleted?$$$$$$

Answered by MJS last updated on 21/Apr/18

$$u=\sqrt{x}$$$$dx=2\sqrt{x}du$$$$2\int u^6{(1-u^2)}^{\frac{3}{2}}du$$$$u=\sin v$$$$du=\cos vdv$$$$2\int \cos v{\sin }^{6}v{(1-{\sin }^{2}v)}^{\frac{3}{2}}dv=$$$$=2\int {\cos }^{4}v{\sin }^{6}vdv=2\int {(1-{\sin }^{2}v)}^2{\sin }^{6}vdv=$$$$=2(\int {\sin }^{10}vdv-2\int {\sin }^{8}vdv+\int {\sin }^{6}vdv)$$$$\mathrm{now}\mathrm{we}\mathrm{have}\mathrm{to}\mathrm{use}$$$$\int {\sin }^n\alpha d\alpha =\frac{n-1}{n}\int {\sin }^{n-2}\alpha d\alpha -\frac{\cos \alpha {\sin }^{n-1}\alpha }{n}$$$$\mathrm{until}{\mathrm{we}}^{\prime }\mathrm{re}\mathrm{done}\mathrm{and}\mathrm{get}$$$$\frac{3v}{128}+\cos v(-\frac{1}{5}{\sin }^{9}v+\frac{11}{40}{\sin }^{7}v-\frac{1}{80}{\sin }^{5}v-\frac{1}{64}{\sin }^{3}v-\frac{3}{128}\sin v)$$$$\mathrm{now}\mathrm{we}\mathrm{go}\mathrm{back}...$$$$v=\arcsin u$$$$\sin v=u$$$$\cos v=\sqrt{1-u^2}$$$$u=\sqrt{x}$$$$\int x^{5/2}{(1-x)}^{3/2}dx=$$$$=\frac{3}{128}\arcsin \sqrt{x}-\frac{\sqrt{x(1-x)}}{640}(128x^4-176x^3+8x^2+10x+15)+C$$

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