(a) (i) Use the substitution sin(u) = x + 3 to show that dx 12x + 7 -1 (x+3) V-2x² – 12x+7+c. sin - |-2x² (ii) In part (i), we could have used the substitution cos(u) = x+3 instead, but the form of the antiderivative would be different. Without doing any calculations, explain how this is accounted for in the final result. 1 (b) In a given situation, the rate of change of the variable x is proportional to V16 – t2" (i) Using k as the constant of proportionality, write down a differential equation that describes this situation. (ii) Find the general solution of the differential equation you wrote down in (b)(i). You will need to make use of a (simple!) trigonometric substitution while solving this equation.

Have you ever observed the movement of the satellites and rockets or how the planets go around the sun in their orbits? How will you determine their motion? Definitely they follow some scientific laws and these kinds of problems are framed as differentia…