Question

What is $\int (x^2+1{)}^{\frac{5}{2}}xdx$ equal to ?
Where c is a constant of integration.

• A. $(x^2+1{)}^{\frac{7}{2}}+c$
• B. $\frac{2}{7}(x^2+1{)}^{\frac{7}{2}}+c$
• C. $\frac{1}{7}(x^2+1{)}^{\frac{7}{2}}+c$
• D. None of the above.

Answer 1

Answer: (C)

$\frac{1}{7}(x^2+1{)}^{\frac{7}{2}}+c$

Let $I=\int {(x^2+1)}^{\frac{5}{2}}xdx$

Let $x^2+1=y$ then, $2xdx=dy$,

Substituting the values, we get

$I=\int (y{)}^{\frac{5}{2}}\frac{dy}{2}$

$=\frac{2y^{\frac{7}{2}}}{2\times 7}+c$

$=\frac{(x^2+1{)}^{\frac{7}{2}}}{7}+c$

Hence, C is correct.