dx-x-2-2x-26-Please-show-your-workings-

Question Number 127203 by naka3546 last updated on 27/Dec/20

$$\int\:\:\frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{26}}}\:\:=\:\:\centerdot\centerdot\centerdot\:\:? \\ $$$${Please}\:\:{show}\:\:{your}\:\:{workings}\:\:! \\ $$

Commented by liberty last updated on 27/Dec/20

∫ (dx/( (√((x+1)^2 +5^2 )))) let x+1 = 5tan q

$$\:\int\:\frac{{dx}}{\:\sqrt{\left({x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} }}\: \\ $$$${let}\:{x}+\mathrm{1}\:=\:\mathrm{5tan}\:{q}\: \\ $$

Commented by Ar Brandon last updated on 27/Dec/20

=Arcsinh(((x+1)/5))+C =ln∣(((x+1)/5))+(√((((x+1)/5))^2 +1))∣+C

$$=\mathrm{Arcsinh}\left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{5}}\right)+\mathcal{C} \\ $$$$=\mathrm{ln}\mid\left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{5}}\right)+\sqrt{\left(\frac{\mathrm{x}+\mathrm{1}}{\mathrm{5}}\right)^{\mathrm{2}} +\mathrm{1}}\mid+\mathcal{C} \\ $$

Answered by Ar Brandon last updated on 27/Dec/20

I=∫(dx/( (√(x^2 +2x+26)))) [t=x+1+(√(x^2 +2x+26)) ⇒ dt=(t/( (√(x^2 +2x+26))))dx] I=∫(1/t)dt=ln∣t∣+C=ln∣(x+1)+(√(x^2 +2x+26)) ∣+C

$$\mathcal{I}=\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}} \\ $$$$\left[\mathrm{t}=\mathrm{x}+\mathrm{1}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}\:\Rightarrow\:\mathrm{dt}=\frac{\mathrm{t}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}}\mathrm{dx}\right] \\ $$$$\mathcal{I}=\int\frac{\mathrm{1}}{\mathrm{t}}\mathrm{dt}=\mathrm{ln}\mid\mathrm{t}\mid+\mathrm{C}=\mathrm{ln}\mid\left(\mathrm{x}+\mathrm{1}\right)+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}\:\mid+\mathrm{C} \\ $$

Answered by Mr.D.N. last updated on 27/Dec/20

∫ (( dx)/( (√(x^2 +2x+26)))) = ∫ (1/( (√(x^2 +2.x.1+1^2 +25))))dx = ∫ (( 1)/( (√((x+1)^2 +(5)^2 ))))dx = log {∣(x+1)+(√((x+1)^2 +5^2 )) ∣ }+c = log (∣x+1+(√(x^2 +2x+26)))∣ +C //.

$$\:\:\int\:\frac{\:\:\mathrm{dx}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}} \\ $$$$\:=\:\int\:\frac{\mathrm{1}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2}.\mathrm{x}.\mathrm{1}+\mathrm{1}^{\mathrm{2}} +\mathrm{25}}}\mathrm{dx} \\ $$$$\:=\:\int\:\frac{\:\:\mathrm{1}}{\:\sqrt{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} +\left(\mathrm{5}\right)^{\mathrm{2}} }}\mathrm{dx} \\ $$$$\:=\:\:\:\mathrm{log}\:\left\{\mid\left(\mathrm{x}+\mathrm{1}\right)+\sqrt{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} }\:\mid\:\right\}+\mathrm{c} \\ $$$$\:=\:\mathrm{log}\:\left(\mid\mathrm{x}+\mathrm{1}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}\right)\mid\:+\mathrm{C}\:\://. \\ $$$$ \\ $$$$ \\ $$$$ \\ $$

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