Sue Pemberton, Julianne Hughes and, Julian Gilbey Solutions for Chapter: Further Calculus, Exercise 9: END-OF-CHAPTER REVIEW EXERCISE 8

Use the substitution method to find the exact value of ${\int }_0^{\frac{1}{4}\pi }\frac{\sqrt{\left(1+3\tan x\right)}}{{\cos }^{2}x} dx$. (Write the answer in fraction.)

The diagram shows part of the curve $y=\left(2x-x^2\right){\mathrm{e}}^{\frac{x}{2}}$ and its maximum point $M.$

Find the exact $x$-coordinate of $M.$

The diagram shows part of the curve $y=\left(2x-x^2\right){\mathrm{e}}^{\frac{x}{2}}$ and its maximum point $M.$

Find the exact value of the area of the shaded region bounded by the curve and the positive $x$-axis.

By first expressing $\frac{4x^2+5x+3}{2x^2+5x+2}$ in partial fractions, show that ${\int }_0^4\frac{4x^2+5x+3}{2x^2+5x+2}dx=8-\ln \left(9\right)$

The diagram shows the curve $y=x^3\ln \left(x\right)$ and its minimum point $M.$

Find the exact coordinates of $M.$

The diagram shows the curve $y=x^3\ln \left(x\right)$ and its minimum point $M.$

Find the exact area of the shaded region bounded by the curve, the $x$-axis and the line $x=2.$

The diagram shows the curve $y=\frac{x^2}{1+x^3}$ for $x⩾0,$ and its maximum point $M.$ The shaded region $R$ is enclosed by the curve, the $x$-axis and the lines $x=1$ and $x=p.$

Find the exact value of the $x$-coordinate of $M.$

The diagram shows the curve $y=\frac{x^2}{1+x^3}$ for $x⩾0,$ and its maximum point $M.$ The shaded region $R$ is enclosed by the curve, the $x$-axis and the lines $x=1$ and $x=p.$

Calculate the value of $p$ for which the area of $R$ is equal to $1.$ Give your answer correct to $3$ significant figures.