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3.6.5 Integration (A-level only)

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Question 13

Given that

5x+3(x−1)(3x+1)2≡Ax−1+B3x+1+C(3x+1)2 \frac{5x+3}{(x-1)(3x+1)^2} \equiv \frac{A}{x-1} + \frac{B}{3x+1} + \frac{C}{(3x+1)^2} (x−1)(3x+1)25x+3​≡x−1A​+3x+1B​+(3x+1)2C​
a.

find the values of the constants AAA, BBB and CCC.

[4]
b.

Hence find the exact value of

∫235x+3(x−1)(3x+1)2 dx \int_{2}^{3} \frac{5x+3}{(x-1)(3x+1)^2} \,\mathrm{d}x ∫23​(x−1)(3x+1)25x+3​dx

giving your answer in the form pln⁡q+rp \ln q + rplnq+r where ppp, qqq and rrr are rational numbers.

[5]

3.6.5 Integration (A-level only) Questions

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