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Question: Integrate: \(\int\limits_{ - 2}^3 {\dfrac{1}{{x + 7}}dx} \)...

Integrate: ∫−231x+7dx\int\limits_{ - 2}^3 {\dfrac{1}{{x + 7}}dx}

Explanation

Solution

Here given an integral of a function of which we have to find the integration of the function, here the function is called the integrand. There are two types of integrals, which are definite integrals and indefinite integrals. Definite integrals are those integrals for which there is a particular limit on the integrals. Whereas for the indefinite integrals there are no such limits on the integral.

Step-By-Step answer:
Definite integrals are integrated on the given limits. These limits have an upper limit and a lower limit where the function or the integrand is integrated from the lower limit to the upper limit of the given limit.
Here a fundamental formula of integration is used to solve the given problem, which is :
⇒∫1xdx=log⁡ex+c\Rightarrow \int {\dfrac{1}{x}dx} = {\log _e}x + c
Where cc is the constant of integration, which is to be included every time we find an integration of a function.
Here in the given integral the function is 1x+7\dfrac{1}{{x + 7}}.
Now we have to solve the integration of ∫−231x+7dx\int\limits_{ - 2}^3 {\dfrac{1}{{x + 7}}dx}
⇒∫−231x+7dx=log⁡e(x+7)−23+c\Rightarrow \int\limits_{ - 2}^3 {\dfrac{1}{{x + 7}}dx} = {\log _e}(x + 7)_{ - 2}^3 + c
⇒∫−231x+7dx=log⁡e(3+7)−log⁡e(−2+7)+c\Rightarrow \int\limits_{ - 2}^3 {\dfrac{1}{{x + 7}}dx} = {\log _e}(3 + 7) - {\log _e}( - 2 + 7) + c
⇒∫−231x+7dx=log⁡e(10)−log⁡e(5)+c\Rightarrow \int\limits_{ - 2}^3 {\dfrac{1}{{x + 7}}dx} = {\log _e}(10) - {\log _e}(5) + c
We know that a basic formula from logarithms is given by log⁡ea−log⁡eb=log⁡e(ab){\log _e}a - {\log _e}b = {\log _e}\left( {\dfrac{a}{b}} \right).
⇒∫−231x+7dx=log⁡e(105)+c\Rightarrow \int\limits_{ - 2}^3 {\dfrac{1}{{x + 7}}dx} = {\log _e}\left( {\dfrac{{10}}{5}} \right) + c
⇒∫−231x+7dx=log⁡e2+c\Rightarrow \int\limits_{ - 2}^3 {\dfrac{1}{{x + 7}}dx} = {\log _e}2 + c

The integration of ∫−231x+7dx=log⁡e2+c\int\limits_{ - 2}^3 {\dfrac{1}{{x + 7}}dx} = {\log _e}2 + c

Note: Note that the definite integrals are generally used to find the areas under a given curve or to find the area between the curves, or to find the area of a function, given the limits of the definite integral. The limits are given in order to find the area of the function within that particular limit, i.e, to integrate the function from the upper limit to the lower limit.