Question

What is the value of integral: $\int {\left( {3x + 5} \right)dx} $ ?

Answer 1

Hint: The given question requires us to integrate a linear function of x with respect to x. Integration gives us a family of curves. Integrals in maths are used to find many useful quantities such as areas, volumes, displacement, etc. integral is always found with respect to some variable, which in this case is x.

Complete step by step solution:
The given question requires us to integrate a linear function given as $\left( {3x + 5} \right)$ in variable x which consists of two terms.
So, we first represent the terms in exponential or power form and then integrate the function directly using the power rule of integration.
So, we can write $\left( {3x + 5} \right)$ as $\left( {3{x^1} + 5{x^0}} \right)$.
Hence, we have to integrate $\left( {3{x^1} + 5{x^0}} \right)$ with respect to x.
So, we have to evaluate $\int {\left( {3{x^1} + 5{x^0}} \right)\,} dx$.
So, first we split the integral into two parts as there are two terms in the linear function of x. So, we get,
So, we get, $\int {\left( {3{x^1} + 5{x^0}} \right)\,} dx = \int {3{x^1}\,} dx + \int {5{x^0}dx} $
$ \Rightarrow \int {\left( {3{x^1} + 5{x^0}} \right)\,} dx = \int {3{x^1}\,} dx + \int {5{x^0}dx} $
Taking constants out of the integral,
$ \Rightarrow \int {\left( {3{x^1} + 5{x^0}} \right)\,} dx = 3\int {{x^1}\,} dx + 5\int {{x^0}dx} $
Now, we know the power rule of integration. According to the power rule of integration, the integral of ${x^n}$ with respect to x is $\dfrac{{{x^{\left( {n + 1} \right)}}}}{{n + 1}}$. So, we get,
$ \Rightarrow \int {\left( {3{x^1} + 5{x^0}} \right)\,} dx = 3\left( {\dfrac{{{x^{1 + 1}}}}{{1 + 1}}} \right) + 5\left( {\dfrac{{{x^{0 + 1}}}}{1}} \right) + c$
$ \Rightarrow \int {\left( {3{x^1} + 5{x^0}} \right)\,} dx = \dfrac{{3{x^2}}}{2} + 5x + c$
Hence, the value of integral $\int {\left( {3x + 5} \right)dx} $ is $\dfrac{{3{x^2}}}{2} + 5x + c$.

Note:
The indefinite integrals of certain functions may have more than one answer in different forms. However, all these forms are correct and interchangeable into one another. Indefinite integral gives us the family of curves as we don’t know the exact value of the constant. The constants multiplied with the terms may be taken out of the integral and power rule of integral $\int {{x^n}dx} = \dfrac{{{x^{\left( {n + 1} \right)}}}}{{n + 1}}$ to solve the problem.