Question

Please help me, I am more than willing to answer questions if you have any :)

Answer 1

Given:

`\int_0^3(x)/(√(x^2+16))dx`

Required:

We need to evaluate the given integral.

Step-by-step explanation:

`Use\text{ 16=4}^2.`

`\int_0^3(x)/(√(x^2+16))dx=\int_0^3(x)/(√(x^2+4^2))dx`
`Let\text{ u=x and dv=}(1)/(√(x^2+4^2))dx.`

`d\text{u=dx and }\int\text{dv=}\int(1)/(√(x^2+4^2))dx.`

`v=In\lvert{(x+√(x^2+4^2))/(4)}\rvert`
`Use\text{ u}\cdot dv=uv-\int vdu.`

`\int_0^3(x)/(√(x^2+16))dx=xIn\lvert{(x+√(x^2+4^2))/(4)}\rvert-\int_0^3In\lvert{(x+√(x^2+4^2))/(4)}\rvert dx`

We know that

`\int_0^3(x)/(√(x^2+16))dx=In\lvert√(x^2+16)+x|+C`
`\int_0^3(x)/(√(x^2+16))dx=In|8|-In|4|`
`\int_0^3(x)/(√(x^2+16))dx=1`

Final answer:

`\int_0^3(x)/(√(x^2+16))dx=1`