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

Express $\frac{5}{x^2+9x+14}$ in partial fractions.

Hence find

$\int \frac{5}{x^2+9x+14}\mathrm{d}x$

expressing your answer as a single logarithm.

Hence show that

${\int }_{-1}^5\frac{5}{x^2+9x+14}\mathrm{d}x=\ln 7-\ln 2$

Write $x^2-10x+29$ in the form ${\left(x-h\right)}^2+k$, where $h,k\in \mathrm{\mathbb{Z}}$.

Hence, using the substitution $u=x-5$, show that

$\int \frac{1}{x^2-10x+29}\mathrm{d}x=\frac{1}{2}\arctan \left(\frac{x-5}{2}\right)+c$

where $c$ is a constant of integration.

Find the exact value of the definite integral

${\int }_5^7\frac{1}{x^2-10x+29}\mathrm{d}x$

Use the substitution $u=1+{\sin }^{3}x$ to show that

${\int }_0^{\frac{\pi }{2}}{\sin }^{2}x\cos x\sqrt{1+{\sin }^{3}x}\mathrm{d}x=\frac{2}{9}\left(2\sqrt{2}-1\right)$

Use integration by parts twice to show that

$\int 32x^2{\mathrm{e}}^{4x}\mathrm{d}x={\mathrm{e}}^{4x}\left(p{x}^2+qx+r\right)+c$

where $p$, $q$ and $r$ are constants to be found, and where $c$ is a constant of integration.

Let $f$ be a function defined for all $x\in \mathrm{ℝ}$. Consider the graph of $y=f\left(x\right)$.

Given that

$\frac{\mathrm{d}y}{\mathrm{d}x}=32x^2{\mathrm{e}}^{4x}$

and that the graph passes through the point $\left(\frac{1}{4},\frac{\mathrm{e}+7}{2}\right)$, find an expression for $f\left(x\right)$.

Let $f$ be the function defined by $f\left(x\right)=cosec2x\cot 2x$, $0<x<\frac{\pi }{2}$.

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 $x$-axis and the line $x=\frac{\pi }{8}$.

Find the exact area of the shaded region.

The following diagram shows a part of the graph of the curve $y=\frac{1}{4}{\left(x-2\right)}^2$ .  The shaded region is the region enclosed by the graph and the positive $x$- and $y$-axes.

(i) Find the coordinates of the points where the graph intersects the coordinate axes.

(ii) For the part of the curve that forms the boundary of the shaded region, show that  $x=2-2\sqrt{y}$ .

Find the area of the shaded region

(i) by calculating it as an area between the curve and the $x$-axis.

(ii) by calculating it as an area between the curve and the $y$-axis.

Find the volume of the solid formed when the shaded region is rotated $2\mathrm{\pi }$ radians about the $x$-axis.

Find the volume of the solid formed when the shaded region is rotated $2\mathrm{\pi }$ radians about the $y$-axis.

The diagram below shows the cross-section of a bowl that a company is planning to begin producing.

As indicated on the diagram, one of the sides of the bowl in the cross-section may be described by the curve $y=\sqrt{x^2-36}$, where units for $x$ and $y$ are centimetres. The cross-section is entirely symmetrical about the $y$-axis. The flat circular bottom of the bowl has a diameter of 12 cm, and the vertical depth of the bowl is 6 cm. For purposes of answering this question, the thickness of the bottom and sides of the bowl may be regarded as negligible.

Find the exact coordinates of the point marked $\mathrm{A}$ on the diagram.

Show that the capacity of the bowl in cm³ is given by

$\pi {\int }_0^b\left(y^2+36\right)\mathrm{d}y$

where $b$ is a constant to be determined.

Hence find the capacity of the bowl.

Find the following indefinite integrals:

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

$\int \frac{\tan 2x}{\cos 2x}\mathrm{d}x$

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

Find the following indefinite integrals:

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

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

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

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

Express $\frac{8}{x^2+2x-15}$ in partial fractions.

Hence show that

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

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

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

Write $x^2+6x+18$ in the form ${\left(x-h\right)}^2+k$, where $h,k\in \mathrm{\mathbb{Z}}$.

Hence find the exact value of the definite integral

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

Write $\frac{u}{2+u}$ in the form $A+\frac{B}{2+u}$, where $A,B\in \mathrm{\mathbb{Z}}$.

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

$\int \frac{3}{2+\sqrt{x+7}}\mathrm{d}x$

Show that

$\int \ln x\mathrm{d}x=x\ln x-x+c$

where $c$ is a constant of integration.

Hence find the exact value of

${\int }_1^{{\mathrm{e}}^3}\ln x\mathrm{d}x$

Consider the integral $I$ defined by

$I=\int {\mathrm{e}}^x\cos x\mathrm{d}x$

Use integration by parts to show that

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

Hence find the indefinite integral

$\int {\mathrm{e}}^x\cos x\mathrm{d}x$

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 $u={\mathrm{e}}^x$, or otherwise, find the exact area of the shaded region.

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.

Show that

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

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

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

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

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 $\mathrm{A}$ 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 $\mathrm{A}$ 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 $2\pi$ radians about the $x$-axis, the resultant solid of revolution has a volume equal to $\frac{1296\pi }{5}$.

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 $R$.

Find the following indefinite integrals:

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

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

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

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

Show that

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

Use the substitution $u^2={\mathrm{e}}^x-1$ to find the exact value of

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

Find the exact value of the definite integral

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

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

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

Find

$\int \arccos x\mathrm{d}x$

Hence find the exact value of

${\int }_{-\frac{1}{2}}^{\frac{\sqrt{3}}{2}}\arccos x\mathrm{d}x$

Show that

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

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

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

Find the value of $a$.

Consider the curve with equation

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

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 $y=\frac{\pi }{6}$ and $y=\frac{\pi }{2}$.

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.

Find the following indefinite integrals:

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

$\int \frac{1}{\sin \frac{x}{4}\tan \frac{x}{4}}\mathrm{d}x$

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

Find the indefinite integral

$\int {\mathrm{e}}^{3x}\sin 2x\mathrm{d}x$

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

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

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

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