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Question Number 62754 by aliesam last updated on 24/Jun/19

$$1)\int \frac{dx}{1-sin\left(x\right)}$$$$$$$$Rsolvein\left(2\right)$$$$$$$${(4-x)}^4+x^4=82$$

Commented by Tony Lin last updated on 25/Jun/19

$$\left(2\right){(4-x)}^4+x^4=81+1=3^4+1^4$$$$f\left(x\right)={(4-x)}^4+x^4-82$$$$f\left(1\right)=0,f\left(3\right)=0$$$$\Rightarrow f\left(x\right)=2(x^4-8x^3+48x^2-128x+87)$$$$=2(x-1)(x-3)(x^2-4x+29)$$$$△={(-4)}^2-4\times 29<0$$$$x^2-4x+29hasnorootsinR$$$$\Rightarrow x=1orx=3$$

Commented by mathmax by abdo last updated on 25/Jun/19

$$letuseanotherwayletI=\int \frac{dx}{1-sinx}changementtan\left(\frac{x}{2}\right)=tgive$$$$I=\int \frac{2dt}{(1+t^2)(1-\frac{2t}{1+t^2})}=\int \frac{2dt}{1+t^2-2t}=\int \frac{2dt}{{(t-1)}^2}=-\frac{2}{t-1}+c$$$$=\frac{2}{1-tan\left(\frac{x}{2}\right)}+c.$$

Answered by Hope last updated on 25/Jun/19

$$1)\int \frac{1+sinx}{{cos}^{2}x}dx$$$$\int {sec}^{2}x+secxtanxdx$$$$tanx+secx+c$$

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