Answer to Question #121895 in Calculus for desmond

Question

$\int_0^1 \frac{1}{\sqrt{16+9x^2}}dx, \int_0^2 \frac{1}{\sqrt{9+4x^2}}dx$

Answer 1

₁₎

$$\int_0^1\frac{1}{\sqrt{16+9x^2}}dx=|t=3x, dt=3dx,t_1=0,t_2=3|=$$

$$\frac{1}{3}\int_0^3\frac{dt}{\sqrt{4^2+t^2}}=\frac{1}{3}ln|t+\sqrt{t^2+16|}|_0^3=ln\sqrt[3]{2}$$

2)

$$\int_0^2\frac{1}{\sqrt{9+4x^2}}dx=|t=2x, dt=2dx,t_1=0,t_2=4|=$$

$$\frac{1}{2}\int_0^4\frac{dt}{\sqrt{3^2+t^2}}=\frac{1}{2}ln|t+\sqrt{t^2+9|}|_0^4=ln\sqrt{3}$$