Question

Solve the indefinite integral (linear):

Answer 1

`\rm \: a. \: \displaystyle \int \rm \: (3 {x}^(2) + 4x)dx`

`= \displaystyle \int \rm 3 {x}^(2) dx + \displaystyle \int \rm4x \: dx`

`\rm = 3 \displaystyle \int \rm {x}^(2) \: dx + 4\displaystyle \int \rm \: x \: dx`

`\rm \: = 3 * \frac{ {x}^(3) }{3} + 4 \frac{ {x}^(2) }{2} + c`

`\boxed{\rm = {x}^(3) + 2 {x}^(2) + c}`

`b. \displaystyle \int \rm \bigg( \frac{1}{ {t}^(2) } + √(t) \bigg)dt`

`= \displaystyle \int \rm \frac{1}{ {t}^(2) } dt + \displaystyle \int \rm √(t) \: dt`

`= \displaystyle \int \rm \: {t}^( - 2) dt \: +\displaystyle \int \rm {t}^{ (1)/(2) } dt`

`\rm \: = \frac{ {t}^( - 1) }{ - 1} + \frac{ {t}^{ (3)/(2) } }{ (3)/(2) } + c \\`

`\rm = - (1)/(t) + (2)/(3) {t}^{ (3)/(2) } + c \\`

[ In Second question I used dt instead of dx ]