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STEP I 1992 question 8 solutionTSR 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. |
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![\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://www.thestudentroom.co.uk/latexrender/pictures/a5/a5211716041831e0e7bbf29ff66a6ef8.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://www.thestudentroom.co.uk/latexrender/pictures/ec/ecc506dc4fd2a07230fc4487413b2b91.png "= \frac{\text{d}}{\text{d} x} [\int f(x) - \int f(a) ]")
![\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://www.thestudentroom.co.uk/latexrender/pictures/f5/f51eec970de8a17a16adb4c7985aaa96.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")
