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Question Number 168499 by infinityaction last updated on 12/Apr/22

Answered by MJS_new last updated on 12/Apr/22

$$\left(1\right)t=\tan \frac{x}{2}$$$$\left(2\right)u=\frac{\sqrt{1-t}}{\sqrt{1+t}}$$$$\Rightarrow$$$$-\int (u^{10}+u^6)du$$$$\mathrm{sorry}{\mathrm{I}}^{\prime }\mathrm{m}\mathrm{too}\mathrm{lazy}\mathrm{to}\mathrm{write}\mathrm{it}\mathrm{all}\mathrm{down}.$$

Commented by infinityaction last updated on 12/Apr/22

$$sirpleaseexplainalittlemore$$

Answered by MJS_new last updated on 12/Apr/22

$$\mathrm{easier}:$$$$\int \frac{{\sec }^{2}x}{{(\sec x+\tan x)}^{9/2}}dx=$$$$[t=\sin x\to dx=\frac{dt}{\cos x}]$$$$=\int \frac{{(1-t)}^{3/4}}{{(1+t)}^{15/4}}dt=$$$$[u=\frac{{(1-t)}^{1/4}}{{(1+t)}^{1/4}}\to dt=-2{(1-t)}^{3/4}{(1+t)}^{5/4}du]$$$$=-\int u^{10}+u^{6}du=-\frac{u^{11}}{11}-\frac{u^7}{7}=$$$$=-\frac{2(9+2t){(1-t)}^{7/4}}{77{(1+t)}^{11/4}}=$$$$=-\frac{2(9+2\sin x){(1-\sin x)}^{7/4}}{77{(1+\sin x)}^{11/4}}+C$$

Commented by peter frank last updated on 13/Apr/22

$$\mathrm{thank}\mathrm{you}$$

Commented by infinityaction last updated on 13/Apr/22

$$thankyousir$$

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