Question

The value of $\int_{}^{}\frac{x^{24}}{x^{10} + 1}$dx ; where t = x⁵.

• A. $\frac { 1 } { 5 }$ $\left\lbrack \frac{t^{3}}{3} - t + \tan^{- 1}t \right\rbrack$
• B. $\left( \frac{t^{3}}{3} - t + \tan^{- 1}t \right)$
• C. $\frac { 1 } { 5 }$ $\left( \frac{t^{3}}{3} + t - \tan^{- 1}t \right)$
• D. $\left( \frac{t^{3}}{3} + t - \tan^{- 1}t \right)$

Answer 1

Answer: (A)

\frac { 1 } { 5 }$$\left\lbrack \frac{t^{3}}{3} - t + \tan^{- 1}t \right\rbrack

Answer 2

$\int \frac { x ^ { 24 } } { x ^ { 10 } + 1 }$dx = $\int_{}^{}\frac{x^{20}.x^{4}}{(x^{5})^{2} + 1}$dt Put x⁵ = t

Ž $\frac { 1 } { 5 }$ $\int_{}^{}\frac{t^{4}}{t^{2} + 1}$dt = $\frac { 1 } { 5 }$ $\int_{}^{}\frac{t^{4} - 1 + 1}{t^{2} + 1}$dt

=$\frac { 1 } { 5 }$ $\left( \frac{t^{3}}{3} - t + \tan^{- 1}t \right)$, t = x⁵