|
|
|
STEP I 1992 question 8 solution
From The Student RoomTSR Wiki > Study Help > Subjects and Revision > Mathematics > STEP > STEP I 1992 question 8 solution
Consider
You can just differentiate both sides from here and get
and plug this into the integrals in the second and third part:
Substitute back in:
Deduce C = Solution by insparato and DFranklin. | 
|
![\frac{\text{d}}{\text{d} x} \displaystyle \int_a^x f(t) \hspace5 \text{d} t = \frac{\text{d}}{\text{d} x}[ \int f(t) \text{ d} t ]_a^x](http://thestudentroom.co.uk/../latexrender/pictures/a06a646db1fc108391f78d438f6c2266.png "\frac{\text{d}}{\text{d} x} \displaystyle \int_a^x f(t) \hspace5 \text{d} t = \frac{\text{d}}{\text{d} x}[ \int f(t) \text{ d} t ]_a^x")
![= \frac{\text{d}}{\text{d} x} [\int f(x) - \int f(a) ]](http://thestudentroom.co.uk/../latexrender/pictures/6d11bb45a77c51105faa13c0cd68b186.png "= \frac{\text{d}}{\text{d} x} [\int f(x) - \int f(a) ]")
.
. As before, differentiate w.r.t. x:
and so 
.
![\displaystyle x-\frac{x^3}{3} = \left[\frac{t^3}{3} - \frac{t^5}{5}\right]_x^1 +x - \frac{x^5}{5} + C](http://thestudentroom.co.uk/../latexrender/pictures/9faf566c9b6fdd21ada6a8d8a7d7b28d.png "\displaystyle x-\frac{x^3}{3} = \left[\frac{t^3}{3} - \frac{t^5}{5}\right]_x^1 +x - \frac{x^5}{5} + C")
.
