Using the substitution $$\(u=\sqrt{x}\)$$, find the exact value of $$\( \int_{3}^{\infty} \frac{1}{(x+1) \sqrt{x}} d x . \)$$ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................ ................................................................................................................................................................
Exam No：9709_w21_qp_31 Year：2021 Question No：4
Answer:
State that \(\frac{\mathrm{d} u}{\mathrm{~d} x}=\frac{1}{2 \sqrt{x}}\) or \(\mathrm{d} u=\frac{1}{2 \sqrt{x}} \mathrm{~d} x\)
Substitute throughout for \(x\) and \(\mathrm{d} x\)
Obtain a correct integral with integrand \(\frac{2}{u^{2}+1}\)
Integrate and obtain term of the form \(k \tan ^{-1} u\)
Use limits \(\sqrt{3}\) and \(\infty\) for \(u\) or equivalent and evaluate trig.
Obtain answer \(\frac{1}{3} \pi\)
Substitute throughout for \(x\) and \(\mathrm{d} x\)
Obtain a correct integral with integrand \(\frac{2}{u^{2}+1}\)
Integrate and obtain term of the form \(k \tan ^{-1} u\)
Use limits \(\sqrt{3}\) and \(\infty\) for \(u\) or equivalent and evaluate trig.
Obtain answer \(\frac{1}{3} \pi\)
Knowledge points:
1.5.3 use the notations to denote the principal values of the inverse trigonometric relations (No specialised knowledge of these functions is required, but understanding of them as examples of inverse functions is expected.)
3.5.1 extend the idea of ‘reverse differentiation’ to include the integration of (Including examples such as.)
3.5.6 use a given substitution to simplify and evaluate either a definite or an indefinite integral.
Solution:
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