Exams Joint Entrance Examination Mathematics Indefinite Integrals Question 26 Question 26 of 416 Medium NUMERICAL
If ∫ 2 x 2 + 5 x + 9 x 2 + x + 1 d x = x x 2 + x + 1 + α x 2 + x + 1 + β log e ∣ x + 1 2 + x 2 + x + 1 ∣ + C \int \frac{2{x}^{2}+5x+9}{\sqrt{{x}^{2}+x+1}}dx=x\sqrt{{x}^{2}+x+1}+\alpha \sqrt{{x}^{2}+x+1}+\beta {\log }_{e}|x+\frac{1}{2}+\sqrt{{x}^{2}+x+1}|+C ∫ x 2 + x + 1 2 x 2 + 5 x + 9 d x = x x 2 + x + 1 + α x 2 + x + 1 + β log e ∣ x + 2 1 + x 2 + x + 1 ∣ + C , where C C C is the constant of integration, then α + 2 β \alpha +2\beta α + 2 β is equal to \_\_\_\_\_\_\_\_\_\_ .
Answer: 16
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2 x 2 + 5 x + 9 = A ( x 2 + x + 1 ) + B ( 2 x + 1 ) + C A = 2 B = 3 2 C = 11 2 \begin{gathered} 2{x}^{2}+5x+9=A({x}^{2}+x+1)+B(2x+1)+C \\ A=2\, B=\frac{3}{2}\, C=\frac{11}{2} \end{gathered} 2 x 2 + 5 x + 9 = A ( x 2 + x + 1 ) + B ( 2 x + 1 ) + C A = 2 B = 2 3 C = 2 11
2 ∫ x 2 + x + 1 d x + 3 2 ∫ 2 x + 1 x 2 + x + 1 d x + 11 2 ∫ d x x 2 + x + 1 2 ∫ ( x + 1 2 ) 2 + ( 3 2 ) 2 d x + 3 x 2 + x + 1 + 11 2 ∫ d x ( x + 1 2 ) 2 + ( 3 2 ) 2 2 ( x + 1 2 2 x 2 + x + 1 + 3 8 l n ( x + 1 2 + x 2 + x + 1 ) ) + 3 x 2 + x + 1 + 11 2 l n ( x + 1 2 + x 2 + x + 1 ) + C α = 7 2 α + 2 β = 16 \begin{gathered} 2\int \sqrt{{x}^{2}+x+1}dx+\frac{3}{2}\int \frac{2x+1}{\sqrt{{x}^{2}+x+1}}dx+\frac{11}{2}\int \frac{dx}{\sqrt{{x}^{2}+x+1}} \\ 2\int \sqrt{{(x+\frac{1}{2})}^{2}+{(\frac{\sqrt{3}}{2})}^{2}}dx+3\sqrt{{x}^{2}+x+1}+\frac{11}{2}\int \frac{dx}{\sqrt{{(x+\frac{1}{2})}^{2}+{(\frac{\sqrt{3}}{2})}^{2}}} \\ 2(\frac{x+\frac{1}{2}}{2}\sqrt{{x}^{2}+x+1}+\frac{3}{8}ln(x+\frac{1}{2}+\sqrt{{x}^{2}+x+1}))+3\sqrt{{x}^{2}+x+1} \\ +\frac{11}{2}ln(x+\frac{1}{2}+\sqrt{{x}^{2}+x+1})+C \\ \alpha =\frac{7}{2} \\ \alpha +2\beta =16 \end{gathered} 2 ∫ x 2 + x + 1 d x + 2 3 ∫ x 2 + x + 1 2 x + 1 d x + 2 11 ∫ x 2 + x + 1 d x 2 ∫ ( x + 2 1 ) 2 + ( 2 3 ) 2 d x + 3 x 2 + x + 1 + 2 11 ∫ ( x + 2 1 ) 2 + ( 2 3 ) 2 d x 2 ( 2 x + 2 1 x 2 + x + 1 + 8 3 l n ( x + 2 1 + x 2 + x + 1 )) + 3 x 2 + x + 1 + 2 11 l n ( x + 2 1 + x 2 + x + 1 ) + C α = 2 7 α + 2 β = 16 Answer the question to unlock