Question

Evaluate the definite integral:
int_(0)^(1) x sqrt(21x^2 + 4) \ dx

If you could show all steps it would be much appreciated.

Answer 1

\[
\int_0^1 x \sqrt{21x^2+4}dx\\
u = 21 x^2 +4,\quad du= 42 x dx,\quad x dx =\frac {du}{42}\\
\int_0^1 x \sqrt{21x^2+4}dx =\frac 1 {42} \int_4^{25} u^{1/2} du
\]
can you finish it from here?

Answer 2

\[
\frac 1 {42} \int u^{1/2} du=\frac 1 {42}\left[ \frac { u^{3/2}}{3/2} \right]=\frac 1 {42} \int_4^{25} u^{1/2} du=\frac{13}{7}
\]