What is $f (x) = \int \frac{3 - x}{x - 4}$ $f(x) = int (3-x)/(x-4)$ if $f (5) = 1$ $f(5)=1$ ?

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Answer 1 · 2017-01-23T03:47:33

$f (x) = 6 - x - ln | x - 4 |$ $f(x) = 6 - x - ln|x- 4|$

$f (x) = \int \frac{- (x - 3)}{x - 4} d x$ $f(x) = int(-(x - 3))/(x- 4)dx$

$f (x) = - \int \frac{x - 3}{x - 4} d x$ $f(x) = -int(x - 3)/(x - 4)dx$

Decompose into partial fractions.

$\frac{A}{1} + \frac{B}{x - 4} = \frac{x - 3}{1 (x - 4)}$ $A/1 + B/(x -4) = (x - 3)/(1(x - 4))$

$A (x - 4) + B (1) = x - 3$ $A(x- 4) + B(1) = x - 3$

$A x - 4 A + B = x - 3$ $Ax - 4A + B = x - 3$

Write a system of equations:

${(A = 1), (B - 4A = -3):}$

Solve:

$A = 1$ and $B = 1$ .

$f(x) =- int 1 + 1/(x - 4)dx$

This can be integrated using $int(1/x) = ln|x| + C$ and $int(x^n)dx = x^(n +1)/(n + 1) + C$ .

$f(x) = -x - ln|x- 4| + C$

Solve for $C$ now.

$1 = -5 - ln|5 - 4| + C$

$1 = -5 + 0 + C$

$C = 6$

The function is therefore $f(x) = 6 - x - ln|x- 4|$ .

Hopefully this helps!

What is f(x) = int (3-x)/(x-4) f(x)=∫3−xx−4f(x) = int (3-x)/(x-4) if f(5)=1 f(5)=1f(5)=1 ?