1. Tardigrade
  2. Question
  3. Mathematics
  4. If g(x) is the inverse of f(x) and f(x) has domain x ∈[1,5], where f(1)=2 and f(5)=10 then the values of ∫ limits15 f(x) d x+∫ limits210 g(y) d y equals

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Q. If $g(x)$ is the inverse of $f(x)$ and $f(x)$ has domain $x \in[1,5]$, where $f(1)=2$ and $f(5)=10$ then the values of $\int\limits_1^5 f(x) d x+\int\limits_2^{10} g(y) d y$ equals

Integrals

Solution:

$y=f(x) \Rightarrow x=f^{-1}(y)=g(y)$
$dy = f ' (x) dx$
$\therefore I=\int\limits_1^5 f(x) d x+\int\limits_1^5 x f^{\prime}(x) d x$
where y is 2 then $x = 1$
y is 10 then $x = 5$
image
$\therefore I =\int\limits_1^5\left(f(x)+x f^{\prime}(x)\right) d x $
$ =\left.x f(x)\right|_1 ^5=5 f(5)-f(1)=5 \cdot 10-2=48$