Question

$\int \frac{x^{24}}{x^{10}+1}dx$ is equal to

• A. $\frac{x^{15}}{15}+\frac{x^5}{5}+\frac{1}{5}{\tan }^{-1}\left(x^5\right)+c$
• B. $\frac{x^{15}}{15}-\frac{x^5}{5}+\frac{1}{5}{\tan }^{-1}\left(x^5\right)+c$
• C. $\frac{x^{14}}{14}-\frac{x^5}{5}+\frac{1}{5}{\tan }^{-1}\left(x^5\right)+c$
• D. $\frac{x^{14}}{14}+\frac{x^5}{5}+\frac{1}{5}{\tan }^{-1}\left(x^5\right)+c$

Answer 1

Answer: (B)

$\frac{x^{15}}{15}-\frac{x^5}{5}+\frac{1}{5}{\tan }^{-1}\left(x^5\right)+c$

$\int \frac{x^{24}}{x^{10}+1}dx$

$=\int \frac{(x^{24}+x^{14}-x^{14}-x^4+x^4}{x^{10}+1})dx$

$=\int (x^{14}-x^4+\frac{x^4}{x^{10}+1})dx$

$=\int x^{14}dx-\int x^{4}dx+\frac{1}{5}\int \frac{5x^4}{(x^5{)}^2+1^2}dx$

$=\frac{x^{15}}{15}-\frac{x^5}{5}+\frac{1}{5}{\tan }^{-1}\left(x^5\right)+c$

Formulas used:

$\int x^{n}dx=\frac{x^{n+1}}{n+1}+c$

$\int \frac{1}{a^2+x^2}dx=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+c$