Question #e9b24

1 Answer
Jan 16, 2018

int\ dx/sqrt(-x^2-8x)=arcsin((x+4)/4)+C

Explanation:

\ \ \ \ \ \ int\ dx/sqrt(-x^2-8x)

Completing the square, we have
=int\ dx/sqrt(16-(x+4)^2)

Substitute u=x+4 and du=dx:
=int\ (du)/sqrt(16-u^2)

Substitute u=4sin(theta) and du=4cos(theta)\ d theta:
=int\ (4cos(theta))/sqrt(16-16sin^2(theta))\ d theta

=int\ (4cos(theta))/(4cos(theta))\ d theta

=int\ d theta

=theta+C

=arcsin(u/4)+C

=arcsin((x+4)/4)+C