Let f(x)=sinh^−1(6x).
Then f(x) can be written in the simplified form (Ax+B)/(sqrt(Cx^2+D).
What are the values of A,B,C and D?
Again, please show me how to do this.
Thanks
Surely you do not mean hyperbolic sin as sinh.
Do you? what do you mean by sinh?
Yes, it is the inverse hyperbolic sin.
asinh(u) = ln(x+sqrt(x^2+1))
That will not simplify into a rational function of x.
Better review these problems a bit.
To find the values of A, B, C, and D in the simplified form of f(x), we need to apply some algebraic manipulation and trigonometric identities. Here's how to do it step by step:
Step 1: Start with the given function f(x) = sinh^−1(6x).
Step 2: Rewrite the sinh^−1(6x) using the definition of sinh^−1 (also known as inverse hyperbolic sine):
sinh^−1(u) = ln(u + sqrt(1 + u^2))
Applying this to f(x), we have:
f(x) = ln(6x + sqrt(1 + (6x)^2))
Step 3: Let's simplify the expression inside the square root using the identity:
1 + u^2 = (1 + u)(1 - u)
Applying this to our square root:
sqrt(1 + (6x)^2) = sqrt((1 + 36x)(1 - 6x))
Step 4: Now, substitute the simplified expression back into f(x):
f(x) = ln(6x + sqrt((1 + 36x)(1 - 6x)))
Step 5: Simplify further by multiplying the terms inside the square root:
f(x) = ln(6x + sqrt(1 + 36x - 6x - 216x^2))
Simplifying the terms:
f(x) = ln(6x + sqrt(-215x^2 + 30x + 1))
Step 6: We want to write f(x) in the form (Ax + B) / sqrt(Cx^2 + D), so let's match the forms by multiplying the numerator and denominator of f(x) by the conjugate of the denominator:
f(x) = ln(6x + sqrt(-215x^2 + 30x + 1)) * (sqrt(Cx^2 + D) / sqrt(Cx^2 + D))
Expanding the denominator:
f(x) = ln((6x)(sqrt(Cx^2 + D)) + sqrt(-215x^2 + 30x + 1)(sqrt(Cx^2 + D)))
Step 7: Equate the original form of f(x) and the expanded form to identify the values of A, B, C, and D:
An equivalent form of f(x) is f(x) = (Ax + B) / sqrt(Cx^2 + D).
Comparing the numerator of both forms, we have:
(Ax + B) = (6x)(sqrt(Cx^2 + D))
Comparing the denominator of both forms, we have:
sqrt(Cx^2 + D) = sqrt(-215x^2 + 30x + 1)
From these equations, we can equate the coefficients and solve for A, B, C, and D.
Note: Due to the complexity of the equation, the values of A, B, C, and D may be difficult to find explicitly. It is recommended to use numerical approximation methods or software tools to get precise values.