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Question Number 113198 by mnjuly1970 last updated on 11/Sep/20

$$....calculus....$$$$\mathcal{E}valuate:::$$$$$$$$\mathrm{I}:={\int }_0^1\frac{1}{\sqrt{x(x+1)(x+2)(x+3)+1}-3x}dx=???$$$$$$$$\text{You can't use 'macro parameter character \#' in math mode}$$$$$$

Answered by MJS_new last updated on 11/Sep/20

$$x(x+1)(x+2)(x+3)+1=$$$$={(x^2+3x+1)}^2$$$$\mathrm{for}0⩽x⩽1:\sqrt{{(x^2+3x+1)}^2}=x^2+3x+1$$$$\Rightarrow \mathrm{the}\mathrm{integral}\mathrm{resolves}\mathrm{to}$$$$\underset{0}{\stackrel{1}{\int }}\frac{dx}{x^2+1}={\left[\arctan x\right]}_0^1=\frac{\pi }{4}$$

Commented by mnjuly1970 last updated on 12/Sep/20

$$thankyousomuch..$$

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