Qiuyun TMUA Simulation Test Paper 2 · Question 17

Consider the following attempt to solve the integral:

R=\int_0^\pi e^{\cos x}\times(\sin^3x)\dd x

I. $u=\cos x,\ \dfrac{\dd u}{\dd x}=-\sin x,\ x=0\Rightarrow u=1,\ x=\pi\Rightarrow u=-1$
II. $R=\displaystyle\int_1^{-1}e^u\times\sin^3x\times\dfrac1{-\sin x}\dd u$
III. $R=\displaystyle\int_1^{-1}-e^u(1-u^2)\dd u$
IV. $R=\displaystyle\int_{-1}^1e^u\dd u-\int_{-1}^1e^uu^2\dd u$
V. $=\left[e^u\right]_{-1}^1-\left[(u^2-2u+2)e^u\right]_{-1}^1$
VI. $=e-\dfrac1e-\left(e-\dfrac1e-\left(2e-\dfrac2e\right)+\left(2e-\dfrac2e\right)\right)=0$

Which one of the following is true?

AThe solution is correct.

BThe solution is wrong; the error arises as a result of (I).

CThe solution is wrong; the error arises as a result of (II).

DThe solution is wrong; the error arises as a result of (III).

EThe solution is wrong; the error arises as a result of (IV).

FThe solution is wrong; the error arises as a result of (V).

GThe solution is wrong; the error arises as a result of (VI).

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