If \[ \int \frac{2x^2 + 5x + 9}{\sqrt{x^2 + x + 1}} \, dx = \sqrt{x^2 + x + 1} + \alpha \sqrt{x^2 + x + 1} + \beta \log_e \left( \left| x + \frac{1}{2} + \sqrt{x^2 + x + 1} \right| \right) + C, \] where \( C \) is the constant of integration, then \( \alpha + 2\beta \) is equal to ________________
First, simplify the integrand by performing a substitution for \( u = x^2 + x + 1 \). This leads to a simpler form for the integral. We integrate and match the terms with the given solution. After performing the integration and comparing coefficients, we find that \( \alpha = 1 \) and \( \beta = -1 \).
Thus,
\[ \alpha + 2\beta = 1 + 2(-1) = 0. \]
Final Answer: \( \alpha + 2\beta = 0 \).
If the system of equations \[ x + 2y - 3z = 2, \quad 2x + \lambda y + 5z = 5, \quad 14x + 3y + \mu z = 33 \]
has infinitely many solutions, then \( \lambda + \mu \) is equal to:
If the equation of the parabola with vertex \( \left( \frac{3}{2}, 3 \right) \) and the directrix \( x + 2y = 0 \) is \[ ax^2 + b y^2 - cxy - 30x - 60y + 225 = 0, \text{ then } \alpha + \beta + \gamma \text{ is equal to:} \]
The observed and normal molar masses of compound MX2 are 65.6 and 164 respectively. The percent degree of ionisation of MX2 is ________________% (Nearest integer).
In Carius method of estimation of halogen, 0.25 g of an organic compound gave 0.15 g of silver bromide (AgBr). The percentage of Bromine in the organic compound is ________________ × 10-1% (Nearest integer).
(Given: Molar mass of Ag is 108 and Br is 80 g mol-1)
For the reaction:
The correct order of set of reagents for the above conversion is :
Match List - I with List - II.