1. (integral) dx / sqrt(9+x^2)
I had a^2 = 9, a=3, x = 3tan(theta), dx = 3(sec(theta))^2, and
sqrt(9+x^2) = 3sec(theta)
I plugged them in and got:
3(sec(theta))^2 / 3(sec(theta)) = (integral) 3sec(theta).
I then integrated it and got ln|sec(theta) + tan(theta)| +c
I just divided the 3 and got: sqrt(9+x^2) / 3
and for tan(theta), i divided the 3 again and got tan(theta)=x/3
So, the ending result I had gotten was :
ln|(sqrt(9+x^2) / 3) + (x/3)| +c But the answer in the book had the same answer, but the 3's were not in. What am I doing wrong??
2.(3/2,0)(integral) dx/sqrt(9- x^2)
I used x = 3sin(theta), a = 3, dx = 3cos(theta), and
sqrt(9- x^2) = 3cos(theta)
I got the integral: 3cos(theta)/3cos(theta) = integral 1 (theta)
I then integrated it and just got theta.
I inversed the sin and got x = 3sin(theta)^-1
My problem is that I plugged the numbers in, and it was undefined. the
answer in the book came out to pi/6