Challenging Integrals for High School Students

Question

I am now in my last year of high school. We have covered all the techniques useful for indefinite integration that are included in our Maths and Further Maths courses. This includes:

• Integration by parts, inspection, substitution, partial fraction decomposition

• Integration of regular and inverse trigonometric, regular and inverse hyperbolic, exponential, logarithmic, polynomial functions

I would like to have some challenging integrals to attack that are possible for me to solve at my current level of knowledge. By challenging, I mean integrals similar to the ones in this document. They were generally enjoyable and very satisfying to solve. If you have an integral that you think I could do that is more challenging than those in the aforementioned link, so much the better.

Thank you for your suggestions.

Answer 1

This one's really nice.

$$\int\frac{1}{x^7-x}\text{ }dx$$

There's a clever trick that can save you from a tedious partial fraction decomposition. Can you find it?

Answer 2

Since you asked for integrals, here is an integral! This one I found quite challenging, but perhaps you'll find a more elegant way of solving it than I did: $$ \int \frac{dx}{(9-36x^2)^{3/2}} \, . $$ NB this definitely can be solved using the methods outlined in your post.

Answer 3

These problems are not challenging but still good to do them

1. $\int\exp(x)\bigg(\frac{1+x\ln x}{x}\bigg)\mathrm dx$

2. $\int \sin(101x) \sin^{99}(x)\mathrm dx$

3. $\int \sqrt{x-\sqrt{x^2-4}} \ \ \mathrm dx$

Play with these they are high school level.

Answer 4

In this question, $a$ is a positive constant.

(i)        Express $\cosh a$ in terms of exponentials.

             By using partial fractions, prove that $$\int_0^1 \frac{1}{x^2 + 2x\cosh a + 1} \mathrm dx = \frac{a}{2 \sinh a} .$$

(ii)       Find, expressing your answers in terms of hyperbolic functions, $$\int_0^\infty \frac{1}{x^2 + 2x \sinh a -1} \mathrm dx$$ and $$\int_0^ \infty \frac{1}{x^4 +2x^2 \cosh a + 1} \mathrm dx.$$

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Find the area of the region between the curve $y = \frac {\ln x}{x} $ and the $x$-axis, for $1 \leq x \leq a$. What happens to this area as $a$ tends to infinity?

Find the volume of the solid obtained when the region between the curve $y = \frac{\ln x}{x}$ and the $x$-axis, for $1 \leq x \leq a$, is rotated through $2π$ radians about the $x$-axis. What happens to this volume as $a$ tends to infinity?

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For $n = 1, 2, 3, ..., $ let

$$I_n = \int_0^1 \frac{t^{n-1}}{(t+1)^n} \mathrm dt. $$

By considering the greatest value taken by $ \frac{t}{t+1} $ for $0 \leq t \leq 1$ show that $I_n+1 < \frac{1}{2}I_n .$

Show also that $I_{n+1} = -\frac{1}{n2^n-1}.$

Deduce that $I_n < \frac{1}{n2^n-1}$

Prove that $$ \ln 2 = \sum_{r=1}^n \frac{1}{r 2^r} + I_{n+1}$$

and hence how that $\frac{2}{3} < \ln 2 < \frac{17}{24}.$

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The above questions are sourced from this website. All credits go to them. If you need any more questions, please just drop a comment.

Answer 5

Not quite an integral, but a differential equation. This one definitely requires some outside of the box thinking!

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Consider the solution $y(x)$ of the differential equation $$ \frac{dy}{dx}=\sqrt{1+y^{2020}} $$ determined by the initial data $y(0)=0$.

Prove that for all $n \in \mathbb{N}$, $y^{(n)}(0)$ is an integer. Then find the first non-zero value of $y^{(n)}(0)$ with $n>1$.

Answer 6

Here are a couple of calculus questions concerning hyperbolic functions:

1. Just as $\sin$ can be defined as the unique function $f:\mathbb{R} \mapsto \mathbb{R}$ satisfying \begin{align} f''(x) &= \color{blue}{-}f(x) \\ f'(x) &= 1 \\ f(x) &= 0 \, , \end{align} $\sinh$ can be defined as the unique function $f:\mathbb{R} \mapsto \mathbb{R}$ satisfying \begin{align} f''(x) &= \color{red}{+}f(x) \\ f'(x) &= 1 \\ f(x) &= 0 \, . \end{align} Derive the exponential form of hyperbolic sine by solving the above equation.
2. The classical way to define $\sinh$ is via the 'unit hyperbola': If the region OPR has an area of $t/2$, then the $x$- and $y$-coordinates are, respectively, $\cosh t$ and $\sinh t$. (If you're interested in the details of how hyperbolic functions can be defined using a hyperbola, then see here.) If however you define $\sinh$ and $\cosh$ using exponentials, then it is possible to prove that the shaded area is equal to $t/2$, given that $OQ = \cosh t$ and $PQ = \sinh t$. Prove this using integration. You may need to use hyperbolic identities to simplify your answer.
3. The graph of $\cosh$ has a curious property. For any interval $[a,b]$, the area under the graph is equal to the length of the arc connecting the points $(a,\cosh a)$ and $(b,\cosh b)$. Prove this using the formula $$ \text{arc length} = \int_{a}^{b}\sqrt{1+\left(f'(x)\right)^2} \, dx \, , $$ where $y=f(x)$ is the graph of the function in question. (Here, $f=\cosh$.) And if you're interested in seeing why the arc length formula works, then see here.

Answer 7

To add yet another answer to your question, here are two integrals which are challenging, but their solutions are not too messy: \begin{align} &\int \sin(\log_2(x)) \, dx \, , \\[5pt] &\int \frac{dx}{\sqrt{e^{2x}+1}} \, . \end{align}

Answer 8

I know I’m late, but still I’d like to contribute:

$$\int\sqrt[3]{\tan x}\,\mathrm dx$$ $$\int\frac{x+\sin x}{1+\cos x}\mathrm dx$$ $$\int\cos^\frac12x+\sec^\frac32x\,\mathrm dx$$ $$\int e^{e^x+e^{-x}}(e^{2x}+2e^x-e^{-x}-1)\mathrm dx$$ $$\int\frac{\mathrm dx}{(x^4+1)^3}$$ $$\int\sqrt{1+\sqrt{1+x}}\,\mathrm dx$$ $$\int\frac{\mathrm dx}{(a+b\sin x)^2}$$ $$\int\frac{\mathrm dx}{\sin x-\sin2x}$$ $$\int\frac{(x^2+1)(x^2+2)}{(x\cos x+\sin x)^4}\mathrm dx$$ $$\int\frac{x^2+n^2-n}{(x\sin x+n\cos x)^2}\mathrm dx$$ $$\int e^x(x^4+2)(x^2+1)^{\frac{-5}2}\mathrm dx$$ $$\int\cot^{-1}(x^2-x+1)\mathrm dx$$ $$\int\frac{\cos9x+\cos6x}{2\cos5x-1}\mathrm dx$$ $$\int\frac{x^2+x}{(e^x+x+1)^2}\mathrm dx$$ $$\int\frac{\cos^2x}{1+\tan x}\mathrm dx$$ $$\int\frac{x^\frac{-7}6-x^\frac56}{\sqrt[3]x\sqrt{x^2+x+1}-\sqrt x\sqrt[3]{x^2+x+1}}\mathrm dx$$ $$\int\frac{(x-1)\sqrt{x^4+2x^3-x^2+2x+1}}{x^2(x+1)}\mathrm dx$$ $$\int\frac{\mathrm dx}{(x^4+1)\sqrt{\sqrt{x^4+1}-x^2}}$$ $$\int\frac{\mathrm dx}{\sqrt{\sin^3x+\sin^4x}}$$ $$\int\left(\frac{3x^2-9x+1}{x^3-x+1}\right)^2\mathrm dx$$ $$\int\frac{p\sin x+q}{a\sin^2x+b\sin x+c}\mathrm dx$$ $$\int\frac{x^6-x^5-x+1}{x^8+x^4+1}\mathrm dx$$ $$\int\frac{\mathrm dx}{(1-x^2)\sqrt[4]{2x^2-1}}$$ $$\int\frac{\sin^2x}{a\sin x+b\cos x}\mathrm dx$$ $$\int\frac{\sqrt[3]{x+a}-\sqrt[3]x}{x^2(\sqrt[3]{x+a}+\sqrt[3]x)}\mathrm dx$$ $$\int\frac{\mathrm dx}{\cot^2x+\cos x}$$ $$\int\frac{x+2}{\sqrt{x^4+8x^3+20x^2+16x+4}}\mathrm dx$$ $$\int\left(\frac{1+x^2}{\sqrt{x^4-x^2+1}}-1\right)\frac{\mathrm dx}{\sqrt{2\sqrt{x^4-x^2+1}+x^2-2}}$$ $$\int\frac{x^2-x+1}{4x^3+4x^2+x}\mathrm dx$$ $$\int\tan x\sqrt{\sin x+1}\,\mathrm dx$$ $$\int\frac{\mathrm dx}{\sqrt[3]{(x+a)^7(x+b)^2}}$$ $$\int\frac{x^2+2x+1+(3x+1)\sqrt{x+\ln x}}{x\sqrt{x+\ln x}\left(x+\sqrt{x+\ln x}\right)}\mathrm dx$$ $$\int\frac{x^3+2}{x^5}\sqrt{(x^3+x^2-1)(2x^3+x^2-2)}\,\mathrm dx$$ $$\int\frac{(2x^{20}+x^{10}-3)\sqrt{x^{10}-1}}{x^{10}(x^{10}+x^6-1)}\mathrm dx$$ $$\int\frac{x\sqrt x-\sqrt x-1}{\sqrt x(x-1)(x+\sqrt x+\sqrt{1-x}-1)}\mathrm dx$$ $$\int\frac{\sqrt{x^4+x}}{2x^3+1}\mathrm dx$$ $$\int\frac{(2x^2+3x)\sqrt[3]{x+1}}{x^6+x^2+2x+1}\mathrm dx$$ $$\int(3x^3+x^2)(x+1)^5\sqrt{-(x^2+x-1)(x^4+3x^2+4x^2+2x+1)}\,\mathrm dx$$ $$\int_0^1\frac{\sin^{-1}x}{x}\mathrm dx$$ $$\int_1^2\frac{\tan^{-1}(x-1)}{x}\mathrm dx$$ $$\int_1^\frac{c}a\frac{\ln x}{ax^2+bx+c}\mathrm dx$$ $$\int_0^\frac\pi2\sin x\sqrt{10-\sin2x}\mathrm dx$$ $$\int_1^e\frac{\ln x}{1+x}\mathrm dx+ \int_1^{\frac1e}\frac{\ln x}{1+x}\mathrm dx$$ $$\int_{0.25}^{0.75}f(f(x))\mathrm dx\text{ where }f(x)=x^3-\frac32x^2+x+\frac14$$ $$\int_0^\frac\pi2\sin x\sin2x\sin3x\sin4x\,\mathrm dx$$ $$\int_0^\infty\frac{\phi^2x+1}{(x^2+2x+2)\sqrt{2x^2+2x+1}}\mathrm dx$$ $$\frac{\int_0^\pi x^3\ln(\sin x)\mathrm dx}{\int_0^\pi x^2\ln(\sqrt 2\sin x)\mathrm dx}$$ $$\int_0^\pi\frac{1-\cos nx}{1-\cos x}\mathrm dx$$ $$\int_0^\infty\frac{(x^2+5)\ln x}{x^4+25}\mathrm dx$$ $$\int_0^\frac\pi2\frac{\mathrm dx}{(a^2\sin^2x+b^2\cos^2x)^2}$$ $$\int_0^\frac12\frac{\tan^{-1}x}{x^2-ax-2a+1}\mathrm dx$$ $$\int_0^\frac12\frac{\ln\left(\frac{1+x}{1-x}\right)}{x^2-ax+2a-1}\mathrm dx$$ $$\text{Evaluate }\int_0^\infty\frac{\mathrm dx}{(x^2+1)^2}\text{ in the fastest way possible.}$$ $$\text{Evaluate }\int_0^1\frac{x^2-3x+1}{x^4+x^2+1}\mathrm dx\text{ without splitting the integral.}$$ $$\int_0^\infty\frac{x\sqrt x-2x+1}{x^3-1}\mathrm dx$$ $$\int_{\frac{25}{169}}^4\sqrt{x^2+1+\sqrt{1+6x^2-3x^4}}\,\mathrm dx$$