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

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.