Let $f(x)+2f\bigl(\tfrac{1}{x}\bigr)=x^2+5$ and $2g(x)-3g\bigl(\tfrac{1}{x}\bigr)=x$, $x>0$. If $\alpha=\int_1^2 f(x)\,dx$ and $\beta=\int_1^2 g(x)\,dx$, then the value of $9\alpha+\beta$ is:

Step 1: Solve for f(x)

We are given the equation $f(x)+2f\bigl(\tfrac{1}{x}\bigr)=x^2+5$. To find $f(x)$, we can substitute $x$ with $\frac{1}{x}$ in the original equation, which gives us $f\bigl(\tfrac{1}{x}\bigr)+2f(x)=\frac{1}{x^2}+5$. We now have a system of two linear equations in terms of $f(x)$ and $f\bigl(\tfrac{1}{x}\bigr)$. Multiplying the second equation by 2 and subtracting the first equation from it eliminates $f\bigl(\tfrac{1}{x}\bigr)$, allowing us to solve for $f(x)$.

Step 2: Calculate $\alpha$

Now that we have an expression for $f(x)$, we can calculate $\alpha$ by integrating $f(x)$ from $1$ to $2$. We integrate each term separately: $\int x^2\,dx = \frac{x^3}{3}$, $\int -\frac{2}{x^2}\,dx = \frac{2}{x}$, and $\int 5\,dx = 5x$. After finding the antiderivative, we evaluate it at the limits of integration.

Step 3: Evaluate $\alpha$

Substitute the limits of integration into the antiderivative. First, substitute $x=2$, then subtract the result of substituting $x=1$. Simplify the expression to find the value of $\alpha$.

Step 4: Solve for g(x)

Similarly, for $g(x)$, we are given $2g(x)-3g\bigl(\tfrac{1}{x}\bigr)=x$. Substitute $x$ with $\frac{1}{x}$ to get $2g\bigl(\tfrac{1}{x}\bigr)-3g(x)=\frac{1}{x}$. We now have a system of two linear equations for $g(x)$ and $g\bigl(\tfrac{1}{x}\bigr)$. Multiply the first equation by 2 and the second by 3, then add them to eliminate $g\bigl(\tfrac{1}{x}\bigr)$ and solve for $g(x)$.

Step 5: Calculate $\beta$

Now we calculate $\beta$ by integrating $g(x)$ from $1$ to $2$. The integral of $2x$ is $x^2$, and the integral of $\frac{3}{x}$ is $3\ln|x|$. We then evaluate this antiderivative at the limits of integration.

Step 6: Evaluate $\beta$

Substitute the limits of integration into the antiderivative for $g(x)$. Remember that $\ln 1 = 0$. Simplify the expression to find the value of $\beta$.

Step 7: Calculate $9\alpha+\beta$

Finally, substitute the calculated values of $\alpha$ and $\beta$ into the expression $9\alpha+\beta$ and simplify to get the final answer.

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