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Question Number 168744 by LEKOUMA last updated on 16/Apr/22

$$Resolve$$$$1)\int \frac{\sqrt{1+\cos x}}{\sin x}dx$$$$2)\int \frac{dx}{1+\sqrt[3]{x+1}}$$$$3)\int \frac{x\tan x}{\cos {}^{4}x}dx$$$$4)\int \frac{dx}{1+\sqrt{x}+\sqrt{1+x}}$$

Answered by bobhans last updated on 17/Apr/22

$$\left(1\right)\int \frac{\sqrt{2}\cos \frac{1}{2}x}{2\sin \frac{1}{2}x\cos \frac{1}{2}x}dx=\frac{\sqrt{2}}{2}\int \frac{dx}{\sin \frac{1}{2}x}$$$$=\sqrt{2}\int \csc \frac{1}{2}xd\left(\frac{1}{2}x\right)=\sqrt{2}\ln \mid \csc \frac{1}{2}x-\cot \frac{1}{2}x\mid +c$$

Answered by bobhans last updated on 17/Apr/22

$$\left(2\right)\int \frac{dx}{1+\sqrt[3]{x+1}};[x=t^3-1]$$$$=\int \frac{3t^{2}dt}{1+t}=3\int \frac{{(t+1)}^2-2t-1}{1+t}dt$$$$=3[\int (t+1)dt-\int \frac{2(t+1)-1}{1+t}dt]$$$$=3[\frac{{(t+1)}^2}{2}-2t+\ln \mid 1+t\mid ]+c$$$$=3[\frac{1+\sqrt[3]{x+1}}{2}-2\sqrt[3]{x+1}+\ln \mid 1+\sqrt[3]{x+1}\mid ]+c$$

Answered by MJS_new last updated on 17/Apr/22

$$\int \frac{dx}{1+\sqrt{x}+\sqrt{x+1}}=$$$$[t=\sqrt{x}+\sqrt{x+1}\to dx=\frac{2\sqrt{x}\sqrt{x+1}}{t}dt;x=\frac{{(t^2-1)}^2}{4t^2}]$$$$=\frac{1}{2}\int \frac{(t-1)(t^2+1)}{t^3}dt=$$$$=\frac{1}{2}\int (1-\frac{1}{t}+\frac{1}{t^2}-\frac{1}{t^3})dt=$$$$=\frac{t}{2}-\frac{1}{2}\ln t-\frac{1}{2t}+\frac{1}{4t^2}=$$$$=\frac{x+2\sqrt{x}-\sqrt{x}\sqrt{x+1}-\ln (\sqrt{x}+\sqrt{x+1})}{2}+C$$

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