Let f(x)+2f(x1)=x2+5 and 2g(x)−3g(x1)=x, x>0. If α=∫12f(x)dx and β=∫12g(x)dx, then the value of 9α+β is:
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Step-by-Step Solution
Step 1: Solve for f(x)
We are given the equation f(x)+2f(x1)=x2+5. To find f(x), we can substitute x with x1 in the original equation, which gives us f(x1)+2f(x)=x21+5. We now have a system of two linear equations in terms of f(x) and f(x1). Multiplying the second equation by 2 and subtracting the first equation from it eliminates f(x1), allowing us to solve for f(x).
Step 2: Calculate α
Now that we have an expression for f(x), we can calculate α by integrating f(x) from 1 to 2. We integrate each term separately: ∫x2dx=3x3, ∫−x22dx=x2, and ∫5dx=5x. After finding the antiderivative, we evaluate it at the limits of integration.
Step 3: Evaluate α
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 α.
Step 4: Solve for g(x)
Similarly, for g(x), we are given 2g(x)−3g(x1)=x. Substitute x with x1 to get 2g(x1)−3g(x)=x1. We now have a system of two linear equations for g(x) and g(x1). Multiply the first equation by 2 and the second by 3, then add them to eliminate g(x1) and solve for g(x).
Step 5: Calculate β
Now we calculate β by integrating g(x) from 1 to 2. The integral of 2x is x2, and the integral of x3 is 3ln∣x∣. We then evaluate this antiderivative at the limits of integration.
Step 6: Evaluate β
Substitute the limits of integration into the antiderivative for g(x). Remember that ln1=0. Simplify the expression to find the value of β.
Step 7: Calculate 9α+β
Finally, substitute the calculated values of α and β into the expression 9α+β and simplify to get the final answer.