Question

How do you find the definite integral of $\dfrac{{dx}}{{x + 2}}$ from $[0,7]$?

Answer 1

Hint: Start by substituting $u = \sin x$. Substitute the values in place of the terms to make the equation easier to solve. Then we will differentiate the term. Now we will substitute these terms in the original expression and integrate.

Complete step-by-step solution:
We know that the integral of the term $\int {\dfrac{1}{x}dx} $ is $\ln x + c$.
Now we will be taking help of the same function to evaluate the integral.
$
   = \int\limits_0^7 {\dfrac{{dx}}{{x + 2}}} \\
   = [\ln (x + 2)_0^7 \\
 $
Now we will evaluate the limits by substituting both the upper limit and lower limit in the integral and then subtracting the lower limit from the upper limit term.
$
   = [\ln (x + 2)_0^7 \\
   = \ln (9) - \ln (2) \\
   = 2\ln (3) - \ln (2) \\
 $
Hence, the integral of $\dfrac{{dx}}{{x + 2}}$ will be $2\ln (3) - \ln (2)$.

Additional Information: A derivative is the rate of change of a function with respect to a variable. Derivatives are fundamental to the solution of problems in calculus and differential equations. In general, scientists observe changing systems to obtain rate of change of some variable of interest, incorporate this information into some differential equation, and use integration techniques to obtain a function that can be used to predict the behaviour of the original system under diverse conditions. The power rule allows us to find the indefinite integrals of a variety of functions like polynomials, functions involving roots, and even some rational functions.

Note: While substituting the terms make sure you are taking into account the degrees and signs of the terms as well. While applying the power rule make sure you have considered the power with their respective signs.