Evaluate the Integral by Using a Substitution Prior to Integration $$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } + 13 } } d x$$

question 127

Multiple Choice

Evaluate the integral by using a substitution prior to integration by parts.
- $$\int \frac { x ^ { 2 } } { \sqrt { x ^ { 2 } + 13 } } d x$$

• A) \(\frac { 3 x } { 2 } \sqrt { x ^ { 2 } + 13 } + \frac { 13 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 13 } \right) + C\)
• B) \(\frac { 3 x } { 2 } \sqrt { x ^ { 2 } + 13 } - \frac { 13 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 13 } \right) + C\)
• C) \(\frac { x } { 2 } \sqrt { x ^ { 2 } + 13 } - \frac { 13 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 13 } \right) + C\)
• D) \(\frac { x } { 2 } \sqrt { x ^ { 2 } + 13 } + \frac { 13 } { 2 } \ln \left( x + \sqrt { x ^ { 2 } + 13 } \right) + C\)

Calculate equilibrium output for firms in a Cournot duopoly setting.

Analyze the effect of cost structures on firms' profitability in a Cournot duopoly.

Apply the Cournot model to various market scenarios and demand functions.

Utilize demand and cost functions to determine firms' profit-maximizing output levels.

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