Question

Which of the following is the indefinite integral of \[2{x^{\dfrac{1}{2}}}\]?
A) \[\dfrac{4}{3}{x^{\dfrac{3}{2}}} + c\]
B) \[\dfrac{2}{3}{x^{\dfrac{3}{2}}} + c\]
C) \[\dfrac{2}{3}{x^{\dfrac{4}{3}}} + c\]
D) None of the above

Answer 1

Hint: We have already studied derivatives. Indefinite integrals are the opposite of derivatives. That is, if we differentiate a function again after integrating it, the question will revert to its original form. The sign of an indefinite integral is \[\int x {\text{ dx}}\]. Some of the basic formulas are given to us. We must apply those formulas in different types of questions. Here in this question, we will use the formula \[\int {{x^n}{\text{ dx}} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c} \].

Complete step-by-step solution:
Given, we have to find the indefinite integral of \[2{x^{\dfrac{1}{2}}}\] which can be written as
\[\int {2{x^{\dfrac{1}{2}}}{\text{ dx}}} \]
We can write this equation as
\[2\int {{x^{\dfrac{1}{2}}}{\text{ dx}}} \]
We took the constant from the integration sign because when a constant is multiplied by x, it remains the same, which is why we can take it and put it in front of the sign. Now, we already know the basic formula, that is
\[\int {{x^n}{\text{ dx}} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c} \]
By comparing we get to know that here,
\[n = \dfrac{1}{2}\]
Now, putting the values in the formula, we get
\[ \Rightarrow 2\int {{x^{\dfrac{1}{2}}}{\text{ dx}}} = 2 \times \dfrac{{{x^{\dfrac{1}{2} + 1}}}}{{\dfrac{1}{2} + 1}} + c\]
Solving further, we will get
\[ \Rightarrow 2\int {{x^{\dfrac{1}{2}}}{\text{ dx}}} = 2 \times \dfrac{{{x^{\dfrac{3}{2}}}}}{{\dfrac{3}{2}}} + c\]
Now taking the denominator above, we get
\[ \Rightarrow 2\int {{x^{\dfrac{1}{2}}}{\text{ dx}}} = 2 \times \dfrac{2}{3} \times {x^{\dfrac{3}{2}}} + c\]
\[ \Rightarrow 2\int {{x^{\dfrac{1}{2}}}{\text{ dx}}} = \dfrac{4}{3}{x^{\dfrac{3}{2}}} + c\]
Hence, the final answer is \[\dfrac{4}{3}{x^{\dfrac{3}{2}}} + c\] and the correct option is A.

Note: Those who are unfamiliar with integration might differentiate the options presented instead of integrating. But keep in mind that you may only use derivatives when you have a limited number of options. Also, don't forget to include c in your solution because c signifies a constant number, hence, it is necessary to include it.