Further Integration (DP IB Analysis & Approaches (AA): HL): Exam Questions

Find the following indefinite integrals:

$\int 3{\mathrm{cosec}}^2\frac{\mathrm{\pi }x}{2}dx$

$\int \frac{\tan 2x}{\cos 2x}dx$

$\int \frac{{\tan }^2\left(x-\frac{\mathrm{\pi }}{3}\right)}{1-{\cos }^2\left(x-\frac{\mathrm{\pi }}{3}\right)}dx$

Find the following indefinite integrals:

$\int \left(\ln 4\right)2^{x}dx$

$\int \frac{3}{9x^2+4}dx$

$\int \frac{2\mathrm{\pi }}{\sqrt{\left(x+5\right)\left(5-x\right)}}dx$

Using a sketch, briefly describe the family of graphs corresponding to all the possible specific solutions to the integral in part (a).

Show that

$\frac{8}{x^2+2x-15}=\frac{A}{x-3}+\frac{B}{x+5}$ 

where A and B are constants to be found.

Hence show that

${\int }_5^8\frac{8}{x^2+2x-15}dx=2\ln 5-\ln 13$

Use the substitution $u=\sin x+1$ to find the exact value of the following definite integral:

${\int }_0^{\frac{\mathrm{\pi }}{6}}\sin 2x{\left(1+\sin x\right)}^{2}dx$

Show that $x^2+6x+18$  may be written in the form $p+{\left(x-q\right)}^2$ ,  where $p$ and $q$ are constants to be determined.

Hence find the exact value of the definite integral

${\int }_{-6}^0\frac{1}{x^2+6x+18}dx$

Show that

 $\frac{u}{2+u}=A+\frac{B}{2+u}$ 

where $A$ and $B$ are constants to be determined.

Use the substitution $u=\sqrt{x+7}$ along with the result from part (a) to find the indefinite integral

$\int \frac{3}{2+\sqrt{x+7}}dx$ 

giving your answer as a function of $x$.

Use integration by parts to show that

$\int \ln xdx=x\ln x-x+c$        

where $c$ is a constant of integration.

Hence find the exact value of the definite integral

 ${\int }_1^{e^3}\ln xdx$ 

being sure to simplify your answer as far as possible.

Show that

$\int ax\cos bxdx=\frac{a}{b}x\sin bx+\frac{a}{b^2}\cos bx+c$ 

where $a,b\in \mathrm{ℝ}$ are non-zero constants, and $c$ is a constant of integration.

A continuous random variable $X$ has the probability density function $f$ given by

 $f\left(x\right)=\{\begin{array}{ll}\frac{x}{\mathrm{\pi }-2}\cos \left(\frac{x}{\sqrt{2}}\right),& 0\le x\le \frac{\mathrm{\pi }}{\sqrt{2}}\\ 0,& \mathrm{otherwise}\end{array}$ 

Find $P\left(0\le X\le \frac{\mathrm{\pi }\sqrt{2}}{6}\right)$.

Consider the integral $I$ defined by

$I=\int e^x\cos xdx$ 

Use integration by parts to show that

$I=e^x\left(\sin x+\cos x\right)-I$

Hence find the indefinite integral

$\int e^x\cos xdx$

Let $f$ be the function defined by 

$f\left(x\right)=\frac{{\mathrm{e}}^x}{1+{\mathrm{e}}^{2x}}$

The diagram below shows a part of the graph of the curve $y=f\left(x\right)$.  The shaded region is the region bounded by the curve, the positive $x$- and $y$-axes and the line $x=\frac{\ln 3}{2}$.

Using the substitution $u=e^x$, or otherwise, find the exact area of the shaded region.

The following diagram shows a part of the graph of the curve $y=kx-x^2$,  where  $k>0$ is a constant.  The point marked A is the vertex of the curve.  Region $R$ is the region enclosed by the curve and the $x$-axis.  Region $S$ is the region enclosed by the curve, the positive $y$-axis, and the line through point with gradient zero.

Show that the part of the curve bordering the region $S$ can also be represented by the curve with equation $x=\frac{1}{2}\left(k-\sqrt{k^2-4y}\right)$ .

When region $R$ is rotated $2\mathrm{\pi }$ radians about the $x$-axis, the resultant solid of revolution has a volume equal to $\frac{1296\mathrm{\pi }}{5}{\mathrm{units}}^3$.

Find the value of $k$.

Use the result from part (a) to find the exact area of region $S$.

Use your answer to part (c) to write down the area of region $R$.

The diagram below shows the cross-section of a miniature goldfish bowl produced by Some Things Fishy, a specialist company supplying products for miniature goldfish enthusiasts.

The glass part of the bowl sits on a solid base, indicated by the shaded region on the diagram.  The cross-section of the glass part of the bowl is symmetrical about the $y$-axis, and may be described by the curve with equation

 $\frac{x^2}{64}+\frac{{\left(y-5\right)}^2}{16}=1$

The dashed horizontal line represents the diameter of the open top of the fishbowl.  All coordinates are expressed in centimetres, and for purposes of answering this question the thickness of the glass sides of the bowl may be regarded as negligible. 

Given that the diameter of the open top of the fishbowl is 15 cm, and that this is less than the diameter of the fishbowl at its widest point, find the capacity of the glass part of the fishbowl.

Find the following indefinite integrals:

$\int 5x^2{\sec }^2\left(2{\mathrm{x}}^3-1\right)\mathrm{d}x$

$\int \frac{1}{\sin \frac{x}{4}\tan \frac{x}{4}}dx$

$\int \left(1+{\cos }^2\left(\frac{x-\mathrm{\pi }}{2}\right)\right)\left(\frac{{\tan }^2\left(\frac{x-\mathrm{\pi }}{2}\right)}{1-{\cos }^4\left(\frac{\mathrm{\pi }-\mathrm{x}}{2}\right)}\right)dx$

Find the following indefinite integrals: 

$\int \frac{1}{12}\left(\ln \frac{1}{64}\right)4^{x}dx$

$\int \frac{3}{11+{\left(x-7\right)}^2}dx$

$\int \frac{e^x}{\sqrt{25-e^{2x}}}dx$

Using a sketch, briefly describe the family of graphs corresponding to all the possible specific solutions to the integral in part (a).

Show that

${\int }_{-8}^{-5}\frac{x+15}{x^2-3x-28}dx=6\ln 2-2\ln 5$

Use the substitution $u^2=e^x-1$ to find the exact value of the following definite integral:

 ${\int }_{\ln 2}^{\ln 3}\frac{e^{4x}}{2e^x-2}dx$ 

being sure to simplify your answer as much as possible.

Find the exact value of the definite integral

${\int }_1^{3+2\sqrt{3}}\frac{2}{\sqrt{7+6x-x^2}}dx$

Use the substitution $u=\sqrt{x^2-5}$ to find the indefinite integral

$\int \frac{x}{3-\sqrt{x^2-5}}dx$

Use integration by parts to find the indefinite integral

$\int \arccos xdx$

Hence find the exact value of the definite integral

 ${\int }_{-\frac{1}{2}}^{\frac{\sqrt{3}}{2}}\arccos xdx$

Show that

$\int a{x}^2\sin bxdx=\left(\frac{2a}{b^3}-\frac{a}{b}{x}^2\right)\cos bx+\frac{2a}{b^2}x\sin bx+c$

where $a,b\in \mathrm{ℝ}$ are non-zero constants, and c is a constant of integration.

A continuous random variable $X$ has the probability density function $f$ given by

$f\left(x\right)=\{\begin{array}{ll}a{x}^2\sin \left(\frac{\pi x}{5}\right),& 0\le x\le 5\\ 0,& \mathrm{otherwise}\end{array}$

Find the value of $a$.

Find the indefinite integral

$\int e^{3x}\sin 2xdx$

Let $f$ be the function defined by $f\left(x\right)=\frac{e^{\frac{1}{2}x}}{1+e^x}$. 

Show that $f$ is an even function.

The diagram shows a part of the graph of $y=f\left(x\right)$.  The shaded region is the region bounded by the curve, the positive $y$-axis, the negative $x$-axis, and the line $x=-\ln 3$.

Find the exact area of the shaded region.

Given that

${\int }_{-\ln 3}^a\frac{e^{\frac{1}{2}x}}{1+e^x}dx=\frac{\mathrm{\pi }}{3}$

write down the value of $a$.  Justify your answer.

Consider the curve with equation

 $x={\mathrm{cosec}}^{n}y\cot y,\frac{\pi }{6}\le y\le \frac{\pi }{2}$ 

where $n\in \mathrm{ℝ},n\ne 0$, .  It is given that, for any allowed value of $n,x$ has a defined non-negative value for all values of $y$ in the stated domain. 

Let $A$ be the area enclosed by the curve, the $y$-axis, and the lines  $y=\frac{\mathrm{\pi }}{6}$  and $y=\frac{\mathrm{\pi }}{2}$ .

 Find an expression for $A$ in terms of $n$.

The diagram below shows the cross-section of a goldfish bowl to be produced by Pieseize Manufacturing, a specialist company supplying products for goldfish enthusiasts.

The glass part of the bowl sits on a solid base, indicated by the shaded region on the diagram. The cross-section of the glass part of the bowl is symmetrical about the $y$-axis, and may be described by the curve with equation

$\frac{x^2}{400}+\frac{{\left(y-17\right)}^2}{225}=1$

The dashed horizontal line represents the diameter of the open top of the fishbowl.  The maximum depth of the fishbowl, measured along the $y$-axis from the diameter of the open top to where the glass part of the bowl meets the base, is indicated by $d$  in the diagram.  All coordinates are expressed in centimetres, and for purposes of answering this question the thickness of the glass sides of the bowl may be regarded as negligible.

The owner of the company, Skodyn Pieseize, is extremely superstitious and is obsessed with the number 23. Therefore he insists that the capacity of the glass part of this new fishbowl must be exactly 23 litres.  Find the value of $d$ that satisfies this requirement.