Evaluate the following definite integral: `int_1^4(x^2+x)/(sqrt(2x+1))dx`

Explanation

By applying Integrating by parts,let` (x^{2}+x)` as the 1st function and `frac{1}{sqrt{2 x+1}}` as the 2nd function, we get
`

int_{2}^{4} frac{(x^{2}+x)}{sqrt{2 x+1}} d x=[(x^{2}+x) cdot sqrt{2 x+1}]_{2}^{4}-int_{2}^{4}(2 x+1) cdot sqrt{2 x+1} d x `
`
=(60-6 sqrt{5})-int_{2}^{4}(2 x+1)^{3 / 2} d x `
`
=(60-6 sqrt{5})-frac{1}{5} cdot[(2 x+1)^{5 / 2}]_{2}^{4} `
`
=(60-6 sqrt{5})-(frac{243}{5}-5 sqrt{5}) `
`
=(frac{57}{5}-sqrt{5})=sqrt{frac{57-5 sqrt{5}}{5}}`
`