Pearson Edexcel IAL Mathematics FP3.4.1 Integration of hyperbolic functions

Practise integrating hyperbolic functions and using identities to turn definite integrals into compact exact logarithmic answers.

Exam points

evaluate ∫coth x dx as ln(sinh x) and simplify ratios with hyperbolic identities
link a curve length integral to a hyperbolic antiderivative before giving an exact log form

[Maximum number: 4]

In this question you must show all stages of your working. Solutions relying on calculator technology are not acceptable.

Figure 1

Figure 1 shows the curve with equation

y=\ln\left(\tanh\frac{x}{2}\right) \quad 1\leqslant x\leqslant 2

Hence show that

s=\ln\left(e+\frac1e\right)

s=\int_1^2\coth x\,\mathrm{d}x

=\left[\ln(\sinh x)\right]_1^2=\ln(\sinh2)-\ln(\sinh1)

=\ln\left(\frac{\sinh2}{\sinh1}\right)

\sinh2=2\sinh1\cosh1\Rightarrow \frac{\sinh2}{\sinh1}=2\cosh1=e+e^{-1}

s=\ln\left(e+\frac1e\right)

B1: integrates coth x to ln(sinh x).
M1/dM1: substitutes limits and uses hyperbolic/exponential identities or log laws.
A1*: obtains the required result by algebraic integration.

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FP3.4.1 - Integration of hyperbolic functions question 1