$\int\limits_{0}^{8}| x -5| d x$ is equal to
- 17
- 9
- 12
- 18
The Correct Option is A
Solution and Explanation
$\Rightarrow x-5=[x-5, x \geq 5$
$[-x+5, x< 5$
$=\int\limits_{0}^{5}(-x+5) d x+\int\limits_{5}^{8}(x-5) d x$
$=\left[\frac{-x^{2}}{2}+5 x\right]_{0}^{5}+\left[\frac{-x^{2}}{2}-5 x\right]_{5}^{8}$
$=\frac{-25}{2}+25+\frac{64}{2}-25-40+25$
$=17$
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Concepts Used:
Methods of Integration
Given below is the list of the different methods of integration that are useful in simplifying integration problems:
Integration by Parts:
If f(x) and g(x) are two functions and their product is to be integrated, then the formula to integrate f(x).g(x) using by parts method is:
∫f(x).g(x) dx = f(x) ∫g(x) dx − ∫(f′(x) [ ∫g(x) dx)]dx + C
Here f(x) is the first function and g(x) is the second function.
Method of Integration Using Partial Fractions:
The formula to integrate rational functions of the form f(x)/g(x) is:
∫[f(x)/g(x)]dx = ∫[p(x)/q(x)]dx + ∫[r(x)/s(x)]dx
where
f(x)/g(x) = p(x)/q(x) + r(x)/s(x) and
g(x) = q(x).s(x)
Integration by Substitution Method
Hence the formula for integration using the substitution method becomes:
∫g(f(x)) dx = ∫g(u)/h(u) du
Integration by Decomposition
Reverse Chain Rule
This method of integration is used when the integration is of the form ∫g'(f(x)) f'(x) dx. In this case, the integral is given by,
∫g'(f(x)) f'(x) dx = g(f(x)) + C