Question and Answers Forum

All Questions      Topic List

None Questions

Previous in All Question      Next in All Question      

Previous in None      Next in None      

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

∫ (dx/( (√(x^2 +2x+26)))) = ∙∙∙ ? Please show your workings !

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

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 //.

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

Terms of Service

Privacy Policy

Contact: info@tinkutara.com