Question

2. Use change of variables technique to integrate the following

Z

x(3x − 2) 1

2 dx

Answer 1

$$\int x(3x-2)^{1/2}dx$$

$$u=3x-2, du=3dx$$

$$x=\dfrac{1}{3}u+\dfrac{2}{3}$$

$$\int x(3x-2)^{1/2}dx=\dfrac{1}{3}\int(\dfrac{1}{3}u+\dfrac{2}{3})u^{1/2}du$$

$$=\dfrac{1}{3}\int(\dfrac{1}{3}u^{3/2}+\dfrac{2}{3}u^{1/2})du$$

$$=\dfrac{1}{3}\bigg(\dfrac{1}{3}(\dfrac{2}{5})u^{5/2}+\dfrac{2}{3}(\dfrac{2}{3})u^{3/2}\bigg)+C$$

$$=\dfrac{2}{45}u^{5/2}+\dfrac{4}{27}u^{3/2}+C$$

$$=\dfrac{2}{45}(3x-2)^{5/2}+\dfrac{4}{27}(3x-2)^{3/2}+C$$

$$=\dfrac{2}{135}(3x-2)^{3/2}(9x-6+10)+C$$

$$=\dfrac{2}{135}(3x-2)^{3/2}(9x+4)+C$$