Question

Calculus - need work shown

Answer 1

`( - 1)/(2) [\cos(8x(x + 3))]+ c`

Answer 2

`\large \text{$ -(1)/(2)\cos (8x^2+24x) + C $}`

Explanation:

`\large \displaystyle\begin{aligned}\textsf{let }\:u & =8x^2+24x\\\\\implies (du)/(dx) & =16x+24\\ & =2(8x+12)\\\\\implies dx & =(1)/(2(8x+12))\: du\\\end{aligned}`

`\large\displaystyle\begin{aligned}\int (8x+12) \sin (8x^2+24x)\:dx & = \int (8x+12) \sin (u) \cdot (1)/(2(8x+12))\:du\\\\& = \int ((8x+12)\sin (u))/(2(8x+12))\:du\\\\& = \int (1)/(2)\sin (u)\:du\\\\& = -(1)/(2)\cos (u) + C\\\\& = -(1)/(2)\cos (8x^2+24x) + C\\\\\end{aligned}`