Question

Find the value of \[\int {(x{x^3}{x^5}{x^7}......} \]to n terms\[)dx\]=
(a) \[{\left( {\dfrac{{{x^{n + 1}}}}{{n + 1}}} \right)^2} + c\]
 (b) \[\dfrac{{{x^{2n + 1}}}}{{2n + 1}} + c\]
(c) \[\dfrac{{{x^{{n^2} + 1}}}}{{{n^2} + 1}} + c\]
(d) \[\dfrac{{{x^{n(n + 1)}}}}{2} + c\]

Answer 1

Hint: Here, we are going to use the properties of exponential powers. Also we can see given power values are in A.P form using the formula of sum n terms of A.P.

Complete step-by-step answer:
Given, \[\int {(x{x^3}{x^5}{x^7}......} to{\text{ }}n{\text{ }}terms)dx\]
\[ \Rightarrow \int {{x^{1 + 3 + 5 + 7......nth}}dx} \]
Here, we can see that 1,3,7,……n are in an A.P, since the common difference is constant and equal to 2.
\[ \Rightarrow \int {{x^{\dfrac{n}{2}(2a + (n - 1)d)}}dx} \](using the formula for sum of n terms given by \[\dfrac{n}{2}\left( {2a + (n - 1)d} \right)\])
Here, we can see that \[a = 1,d = 2\]
\[ \Rightarrow \int {{x^{\dfrac{n}{2}(2 \times 1 + (n - 1)2)}}dx} \]
\[ \Rightarrow \int {{x^{\dfrac{n}{2}(2 + 2n - 2)}}dx} \]
\[ \Rightarrow \int {{x^{{n^2}}}dx} \](using the rule which is given by \[\int {{x^n}dx} = \dfrac{{{x^{n + 1}}}}{{n + 1}} + c\])
\[ \Rightarrow \dfrac{{{x^{{n^2} + 1}}}}{{{n^2} + 1}} + c\]
Therefore, option (c) \[\dfrac{{{x^{{n^2} + 1}}}}{{{n^2} + 1}} + c\] is the required solution

Note: Since, the exponential power had the sum of the A.P term wise. Therefore, we could use the formula for an A.P in general.