Further Integration (DP IB Applications & Interpretation (AI): HL): Exam Questions

Integrate

$\int \cos 2xdx$

Show that

$\int (3x-1{)}^{3}dx=\frac{1}{12}(3x-1{)}^4+c$

where $c$ is a constant of integration.

Find an expression for $y$ given that

$\frac{dy}{dx}={e^5}^x$

and that $y=1$   when $x=0$ .

Find the indefinite integral for

$\int \left(\sqrt{x}+\frac{3}{\sqrt{x}}\right)dx$

Find the indefinite integral for

$\int \frac{x^{{\displaystyle \frac{2}{3}}}+x^{{\displaystyle \frac{11}{6}}}}{x^2}dx$ 

Given that $f\left(x\right)=2x^3+4x$, find $f\text{'}\left(x\right)$.

Hence, or otherwise, find

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

Consider the function $f\left(x\right)=\ln \left(2x^2+1\right)$.

Find $f\text{'}\left(x\right)$.

Hence, find

$\int \frac{x}{2x^2+1}dx$

Let $f\text{'}\left(x\right)=x^2\cos \left(x^3+1\right)$.

Find $f\left(x\right)$, given that $f\left(-1\right)=1$.

Show that

$\frac{\tan x}{\sin x\cos x}=\frac{1}{{\cos }^{2}x}$

Hence find

$\int \frac{3\tan x}{5\sin x\cos x}\mathrm{d}x$

The diagram below shows the graphs of the line $y=6-x$ and the curve $y=x^2.$

Point $P$ is the point of intersection of the curve  $y=x^2$  with the ­$x$ ­-axis.  Point $Q$  is the point of intersection of the curve $y=x^2$ with the line $y=6-x$for which $x>0$.  Point $R$ is the point of intersection of the line  $y=6-x$  with the $x$-axis.

Write down the $x$-coordinates of points $P,Q\mathrm{and}R.$

Calculate the area of the shaded region.

The diagram below shows the graph of the function $f$ which is defined by

$f\left(x\right)=x\sin x,0\le x\le 2\pi$

The shaded region in the diagram is the region enclosed by the $x$-axis and the graph of $y=f\left(x\right)$.

Find the area of the part of the shaded region that lies above the $x$-axis.

Find the area of the entire shaded region.

The diagram below depicts the design for a new company logo. The logo is formed by a circle centred on the origin, which is divided into two regions by the curve $y=f\left(x\right)$ where $f$ is the function defined by $f\left(x\right)=\cos \frac{3x}{2},-\pi \le x\le \pi$. The points where the circle and the curve intersect lie on the $x$-axis, as shown.

The shaded region in the diagram is the region inside that circle that lies below the curve $y=f\left(x\right).$ .

(i) Write down the radius of the circle that forms the outer border of the logo.  

(ii) Hence determine the exact area of the shaded region.

Find the percentage of the circular logo that is shaded.

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

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

where $b$ is a constant to be determined.

Hence find the capacity of the bowl.

Find the indefinite integral for

$\int \sin \left(\frac{\sqrt{3}}{2}x\right)dx$

Find the indefinite integral for

$\int \frac{7}{e^{4x-9}}dx$

Find an expression for $y$ given that

$\frac{dy}{dx}=\cos \left(2\left(\frac{\mathrm{\pi }}{8}-x\right)\right)$

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.

Consider the function  $f$ defined by $f\left(x\right)=\left(x^2-x-2\right)\left(x-5\right),-2\le x\le 4.$

Find the coordinates of the points where the graph of  $y=f\left(x\right)$ intercepts the $x$-axis.

Find the indefinite integral

$\int \left(x^2-x-2\right)\left(x-5\right)dx$

Use your answer to part (b) to calculate the area of the region enclosed by the graph of  $y=f\left(x\right)$ and the $x$-axis.

Find the indefinite integral

$\int \frac{3}{2x}dx$

Find an expression for given that

 $\frac{dy}{dx}=e^{2x+3}+2$

 and also that $y=5$ when $x=-\frac{3}{2}$ .

Find the indefinite integral for

$\int \left(\frac{1}{4}\sqrt{x}-\frac{2}{\sqrt[3]{x}}\right)dx$

Find the indefinite integral for

$\int \frac{x\frac{5}{6}-5x\frac{7}{3}}{x\sqrt{x}}dx$

Consider the function $f\left(x\right)=\ln \left(3x^2-12x+1\right)$.

(i) Find $f\text{'}\left(x\right)$.

(ii) Hence, find

$\int \frac{16-8x}{3x^2-12x+1}\mathrm{d}x$

Let $g\text{'}\left(x\right)=\left(x^2-5x+6\right)\sin \left(2x^3-15x^2+36x-\frac{\pi }{3}\right)$.

Find $g\left(x\right)$ given that $g\left(0\right)=1$.

Work out the following indefinite integrals:

$\int \frac{2}{3{\cos }^{2}3x}dx$

$\int \frac{7\tan \left(\frac{x}{2}\right)}{6\sin \frac{x}{2}\cos \frac{x}{2}}dx$

Find the value of

${\int }_2^5\frac{x+1}{x^2+2x-5}\mathrm{d}x$

giving your answer in the form $a\ln b$, where $a,b\in \mathrm{\mathbb{Q}}$.

The diagram below shows the graph of the function $f$ which is defined by

$f\left(x\right)=-x\cos x,0\le x\le \frac{3\mathrm{\pi }}{2}$

The shaded region in the diagram is the region enclosed by the $x$-axis and the graph of $y=f\left(x\right)$. 

Explain why the area of the shaded region is not equal to

${\int }_0^{\frac{3\mathrm{\pi }}{2}}-x\cos xdx$

Find the area of the entire shaded region.

Find the individual areas of the parts of the shaded region (i) above and (ii) below the x-axis.

The shaded region in the diagram below depicts the design for a new company logo.  The upper border of the logo is formed by a part of the curve with equation $y=\frac{9{\mathrm{\pi }}^2-16x^2}{32}$,  while the lower border of the logo is formed by a part of the curve with equation  $y=-1+2{\sin }^{2}x$.  The points where the two curves intersect lie on the $x$-axis, as shown.

Find the exact coordinates of 

(i) the points of intersection of the two curves 

(ii) the other points where the lower border of the logo intersects the $x$-axis.

Hence find the area of the company logo.

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 A with gradient zero.

Show that the part of the curve bordering the region  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 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 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.

Consider the function $f$ defined by $f\left(x\right)=\left(x^2-3x+2\right)\left(x+2\right),-3\le x\le \frac{5}{4}.$

Find the indefinite integral

$\int \left(x^2-3x+2\right)\left(x+2\right)dx$

Use your answer to part (a) to calculate the area of the region enclosed by the graph of  $y=f\left(x\right)$ and the $x$-axis.

Find the indefinite integral

$\int -\frac{7}{5x}dx$

Find an expression for given that

$\frac{dy}{dx}=x{e}^{x^2-2}$

and also that $y=3$ when $x=-\sqrt{2}$

Find the indefinite integral for

$\int \left(\sqrt{3x}+\frac{5}{\sqrt[3]{8x}}\right)dx$

Find the indefinite integral for

$\int \frac{{\left(8x\right)}^{\frac{2}{3}}-5x^{\frac{1}{6}}}{x\sqrt[3]{x}}dx$

Find the indefinite integral

$\int \frac{x+2x^3}{\left(1-x^2\right)\left(2+x^2\right)}dx$

Let  $g\text{'}\left(x\right)=\frac{\cos \left(\ln x\right)}{x}$ for $x>0.$ 

Find  $g\left(x\right)$ given that  $g\left(1\right)=\mathrm{\pi }$.

Show that

$\int \tan xdx=\ln \left|\frac{1}{\cos x}\right|+c$

A curve with equation $y=f\left(x\right)$ is such that

 $\frac{dy}{dx}=\frac{k}{\left(1+\sin \pi x\right)\left(1-\sin \pi x\right)}$ 

where $k$ is a real constant. 

Given that the curve passes through the points $\left(0,-3\right)$ and $\left(-\frac{1}{4},-\mathrm{\pi }\right)$, find $f\left(x\right)$.

Explain why

$\frac{1}{\tan \theta }=\frac{\cos \theta }{\sin \theta }$

Use the result from part (a) to show that

${\int }_0^{\sqrt{\frac{\pi }{6}}}\frac{x}{\tan \left(x^2-\frac{2\pi }{3}\right)}\mathrm{d}x=-\frac{1}{2}\ln \left(\frac{\sqrt{3}}{2}\right)$

Using your knowledge of the natural logarithm function, explain (without using your GDC) why the value of the integral found in part (b) is a positive number.

The diagram below shows the graph of the function $f$ which is defined by

$f\left(x\right)=x\sin 2x,0\le x\le \frac{3\mathrm{\pi }}{2}$

The shaded region in the diagram is the region enclosed by the $x$-axis and the graph of $y=f\left(x\right)$.  The three sub-parts of the shaded region are denoted by R, S and T, as shown.

Find the value of

${\int }_0^{\frac{3\mathrm{\pi }}{2}}x\sin 2xdx$

Find the individual areas of each of the three sub-parts R, S and T of the shaded region.

Compare the sum of the answers in part (b) to the answer in part (a) and comment on the result.

The diagram below depicts the design for a new company logo.  The upper border of the logo is formed by a part of the curve with equation  $y=9-x^2$,  while the lower border of the logo is formed by a part of the curve with equation $y=\frac{1}{8}{x}^2-5$.  As shown in the diagram, the logo is divided into a shaded part and an unshaded part by a part of the curve with equation $y=\frac{\left(9-x^2\right)\left(2x-1\right)}{10}$.

Find the percentage of the total area of the logo that is shaded.

The following diagram shows a part of the graph of the curve with equation $y^2=k\left(1600-25{\left(x-8\right)}^2\right)$,  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, for the part of the curve where $y$ is non-negative.  Region $S$ is the region enclosed by the curve, the positive $y$-axis, and the line through point A with gradient zero.

When region $R$ is rotated $2\mathrm{\pi }$ radians about the $x$-axis, the resultant solid of revolution has a volume equal to $\frac{12800\mathrm{\pi }}{3}{\mathrm{units}}^3$. 

Find the area of region $S$ by calculating an area between the curve and the $y$-axis.

Find the area of region S by calculating an area between the curve and the $x$-axis.  Confirm that this matches your answer to part (a).