Question

What is the integral of $f\left( x \right)g\left( x \right)$?

Answer 1

Hint: From the question given we have to find the integral of $f\left( x \right)g\left( x \right)$. This can be integrated by using integration by parts rule. So, we will first find the integral of the derivative of the first function times the integral of the second function and then subtract it from the product of the first function and integral of the second function.

Complete step by step solution:
Integration by Parts Rule: If the integrand function can be represented as a multiple of two or more functions, the integration of any given function can be done by using the Integration by Parts rule. Let us take an integrand function which is equal to $u\left( x \right)v\left( x \right)$. In mathematics, Integration by parts basically uses the ILATE rule that helps to select the first function and second function in the integration by parts method. Integration by parts formula,
$\Rightarrow \int{u\left( x \right)v\left( x \right)}.dx=u\left( x \right).\int{v\left( x \right).dx}-\int{\left( {{u}^{'}}\left( x \right).\int{v\left( x \right).dx} \right).dx}$
The integration by parts formula can be further written as integral of the product of any two functions = (First function $\times $ Integral of the second function) - Integral of [ (differentiation of the first function) Integral of the second function)
Ilate Rule: In integration by parts, we have learned that when the product of two functions is given to us then we apply the required formula. The integral of the two functions is taken by considering the left term as first function and second term as the second function. This method is called lLATE rule. Suppose, we have to integrate $x{{e}^{x}}$, then we consider x as the first function and ${{e}^{x}}$ as the second function So basically, the first function is chosen in such a way that the derivative of the function could be easily integrated. Usually, the preference order of this rule is in descending order on some functions such as inverse, Logarithm, Algebraic, Trigonometric, Exponent. This rule helps us to solve integration by parts examples using integration by parts formula. Ilate is a rule which helps to decide which term should you differentiate first and which term should you integrate first. ILATE abbreviation is I- inverse, L-logarithmic, A-algebraic, T-trigonometric, E-exponential. The term which is closer to I differentiated first and the term which is closer to E is integrated first.
Now, back to our question the integral of
$\Rightarrow f\left( x \right)g\left( x \right)$
Based on integration by parts rule and using ILATE rule
Let $f\left( x \right)$is a first function and $g\left( x \right)$is a second function based on ilate rule, then by using integration by parts rule formula,
$\Rightarrow \int{f\left( x \right)g\left( x \right)}.dx=f\left( x \right).\int{g\left( x \right).dx}-\int{\left( {{f}^{'}}\left( x \right).\int{g\left( x \right).dx} \right).dx}$

Note: Students should know that we do not add any constant while finding the integral of the second function. Usually, if any function is a power of x or a polynomial in x then we take it as the first function. However, in cases where another function is an inverse trigonometric function or logarithmic function then we take them as the first function. If in the above case if we take $g\left( x \right)$is a first function and $f\left( x \right)$is a second function based on ilate rule, then the integration by using by parts will be,
$\Rightarrow \int{f\left( x \right)g\left( x \right)}.dx=g\left( x \right).\int{f\left( x \right).dx}-\int{\left( {{g}^{'}}\left( x \right).\int{f\left( x \right).dx} \right).dx}$