The diagram shows the curve \( y=x \mathrm{e}^{-\frac{1}{4} x^{2}} \), for \( x \geqslant 0 \), and its ma Using the substitution \( x=\sqrt{u} \), or otherwise, find by integration the ex. bounded by the curve, the \( x \)-axis and the line \( x=3 \).
2. Let \( \mathrm{f}(x)=\frac{4-x+x^{2}}{(1+x)\left(2+x^{2}\right)} \).
(a) Express \( \mathrm{f}(x) \) in partial fractions.
(b) Find the exact value of \( \int_{0}^{4} f(x) d x \). Give your answer as a
3. Find the exact value of \( \int_{0}^{\frac{1}{4} \pi} x \sec ^{2} x \mathrm{~d} x \).
4. Let \( f(x)=\frac{1}{(9-x) \sqrt{x}} \).
Using the substitution \( u=\sqrt{x} \), show that \( \int_{0}^{4} f(x) d x=\frac{1}{3} \ln \).
5. Using the substitution \( u=\sqrt{x} \), find the exact value of
\[
\int_{3}^{\infty} \frac{1}{(x+1) \sqrt{x}} d x \text {. }
\]
