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Question Number 113418 by bobhans last updated on 13/Sep/20

 ∫ ((3x−2)/( (√(x^2 +2x+26)))) dx

$$\:\int\:\frac{\mathrm{3x}−\mathrm{2}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}}\:\mathrm{dx} \\ $$

Commented by bobhans last updated on 13/Sep/20

great all answer..

$$\mathrm{great}\:\mathrm{all}\:\mathrm{answer}.. \\ $$

Answered by Dwaipayan Shikari last updated on 13/Sep/20

(3/2)∫((2x−(4/3))/( (√(x^2 +2x+26))))dx  =(3/2)∫((2x+2)/( (√(x^2 +2x+26))))−(3/2)∫(((10)/3)/( (√(x^2 +2x+26))))  =(3/2)∫(dt/( (√t)))−5∫(1/( (√((x+1)^2 +5^2 ))))dx  =3(√t)−5∫((cosθdθ)/( (√((5sinθ)^2 +5^2 ))))              x+1=sinθ ,1=cosθ(dθ/dx)  =3(√(x^2 +2x+26))−∫((cosθdθ)/( (√(sin^2 θ+1))))  =3(√(x^2 +2x+26)) −∫(du/( (√(u^2 +1))))  =3(√(x^2 +2x+26))−log(u+(√(u^2 +1)))+C  =3(√(x^2 +2x+26))−5log(x+1+(√(x^2 +2x+2)))+C

$$\frac{\mathrm{3}}{\mathrm{2}}\int\frac{\mathrm{2}{x}−\frac{\mathrm{4}}{\mathrm{3}}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{26}}}{dx} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}\int\frac{\mathrm{2}{x}+\mathrm{2}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{26}}}−\frac{\mathrm{3}}{\mathrm{2}}\int\frac{\frac{\mathrm{10}}{\mathrm{3}}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{26}}} \\ $$$$=\frac{\mathrm{3}}{\mathrm{2}}\int\frac{{dt}}{\:\sqrt{{t}}}−\mathrm{5}\int\frac{\mathrm{1}}{\:\sqrt{\left({x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} }}{dx} \\ $$$$=\mathrm{3}\sqrt{{t}}−\mathrm{5}\int\frac{{cos}\theta{d}\theta}{\:\sqrt{\left(\mathrm{5}{sin}\theta\right)^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} }}\:\:\:\:\:\:\:\:\:\:\:\:\:\:{x}+\mathrm{1}={sin}\theta\:,\mathrm{1}={cos}\theta\frac{{d}\theta}{{dx}} \\ $$$$=\mathrm{3}\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{26}}−\int\frac{{cos}\theta{d}\theta}{\:\sqrt{{sin}^{\mathrm{2}} \theta+\mathrm{1}}} \\ $$$$=\mathrm{3}\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{26}}\:−\int\frac{{du}}{\:\sqrt{{u}^{\mathrm{2}} +\mathrm{1}}} \\ $$$$=\mathrm{3}\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{26}}−{log}\left({u}+\sqrt{{u}^{\mathrm{2}} +\mathrm{1}}\right)+{C} \\ $$$$=\mathrm{3}\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{26}}−\mathrm{5}{log}\left({x}+\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{2}{x}+\mathrm{2}}\right)+{C} \\ $$

Answered by Khalmohmmad last updated on 13/Sep/20

Answered by 1549442205PVT last updated on 13/Sep/20

 F=∫ ((3x−2)/( (√(x^2 +2x+26))))=(3/2)∫((2x+2)/( (√(x^2 +2x+26))))dx  −5∫(dx/( (√(x^2 +2x+26)))).Put x^2 +2x+26=u  ⇒du=(2x+2)dx  (3/2)∫((2x+2)/( (√(x^2 +2x+26))))dx=(3/2)∫(du/( (√u)))=3(√u)  =3(√(x^2 +2x+26))  ∫(dx/( (√(x^2 +2x+26))))=∫((d(x+1))/( (√((x+1)^2 +25))))  =ln∣x+1+(√(x^2 +2x+26))∣     (Since  ∫(dx/( (√(x^2 +λ))))=ln∣x+(√(x^2 +λ)) ∣)  Thus,F=3(√(x^2 +2x+26))−5ln∣x+1+(√(x^2 +2x+26))∣+C

$$\:\mathrm{F}=\int\:\frac{\mathrm{3x}−\mathrm{2}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}}=\frac{\mathrm{3}}{\mathrm{2}}\int\frac{\mathrm{2x}+\mathrm{2}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}}\mathrm{dx} \\ $$$$−\mathrm{5}\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}}.\mathrm{Put}\:\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}=\mathrm{u} \\ $$$$\Rightarrow\mathrm{du}=\left(\mathrm{2x}+\mathrm{2}\right)\mathrm{dx} \\ $$$$\frac{\mathrm{3}}{\mathrm{2}}\int\frac{\mathrm{2x}+\mathrm{2}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}}\mathrm{dx}=\frac{\mathrm{3}}{\mathrm{2}}\int\frac{\mathrm{du}}{\:\sqrt{\mathrm{u}}}=\mathrm{3}\sqrt{\mathrm{u}} \\ $$$$=\mathrm{3}\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}} \\ $$$$\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}}=\int\frac{\mathrm{d}\left(\mathrm{x}+\mathrm{1}\right)}{\:\sqrt{\left(\mathrm{x}+\mathrm{1}\right)^{\mathrm{2}} +\mathrm{25}}} \\ $$$$=\mathrm{ln}\mid\mathrm{x}+\mathrm{1}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{2x}+\mathrm{26}}\mid\:\:\:\:\:\left(\mathrm{Since}\right. \\ $$$$\left.\int\frac{\mathrm{dx}}{\:\sqrt{\mathrm{x}^{\mathrm{2}} +\lambda}}=\mathrm{ln}\mid\mathrm{x}+\sqrt{\mathrm{x}^{\mathrm{2}} +\lambda}\:\mid\right) \\ $$$$\mathrm{Thu}\boldsymbol{\mathrm{s}},\boldsymbol{\mathrm{F}}=\mathrm{3}\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{\mathrm{x}}+\mathrm{26}}−\mathrm{5l}\boldsymbol{\mathrm{n}}\mid\boldsymbol{\mathrm{x}}+\mathrm{1}+\sqrt{\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\mathrm{2}\boldsymbol{\mathrm{x}}+\mathrm{26}}\mid+\boldsymbol{\mathrm{C}} \\ $$

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