Find $\displaystyle\displaystyle \displaystyle\int_{0}^{4} \mathrm{e}^{|x-1|} \, \mathrm{d}x$, lea

Question

Find $\displaystyle\displaystyle \displaystyle\int_{0}^{4} \mathrm{e}^{|x-1|} \, \mathrm{d}x$, leaving your answer in exact form.[4] Find $\displaystyle\displaystyle \displaystyle\int \sin^{2} 3x + \tan^{2} 3x \, \mathrm{d}x$.[3] Use the substitution $x = u^{6}$, where $u > 0$, to find $\displaystyle\displaystyle \displaystyle\int \frac{1}{\sqrt{x} + \sqrt[3]{x}} \, \mathrm{d}x$.[5]

Accepted Answer

(a) $e^3+e-2$; (b) $-\frac1{12}\sin6x+\frac13\tan3x-\frac12x+C$; (c) $2\sqrt{x}-3\sqrt[3]{x}+6\sqrt[6]{x}-6\ln(\sqrt[6]{x}+1)+C$.

2025 EJC P1 Q10

2025 EJC P1 Q10

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