Articles of integration

Integration of a function with respect to another function.

What is the intuition/idea behind integration of a function with respect to another function? Say $$\int f(x)d(g(x)) \;\;\;\;\;?$$ or may be a more particular example $$\int x^2d(x^3)$$ My concern is not at the level of problem solving. To solve we could simply substitute $u=x^3$ and then $x^2=u^{2/3}$. My concern is rather about what meaning physical/geometrical […]

Evaluate $ \int_{25\pi/4}^{53\pi/4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx $

How to evaluate the integral $$\int_{25\pi / 4}^{53\pi / 4}\frac{1}{(1+2^{\sin x})(1+2^{\cos x})}dx\ ?$$

Prove that $\lim_{h \to 0}\frac{1}{h}\int_0^h{\cos{\frac{1}{t}}dt} = 0$

I’m trying to prove that $$\lim_{h \to 0}\frac{1}{h}\int_0^h{f(t)dt} = 0$$ where $$f(t) = \begin{cases}\cos{\frac{1}{t}} &\text{ if } t \neq 0\\ 0&\text{otherwise}\end{cases}.$$ Can someone give me a hint where to start? Darboux sums somehow seem to lead me nowhere. NOTE: I cannot assume that $f$ has an antiderivative $F$.

Proving that $\int_0^\infty\Big(\sqrt{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$

How can we prove, without employing the aid of residues or various transforms, that, for $n>2$ $$\int_0^\infty\Big(\sqrt[n]{1+x^n}-x\Big)dx~=~\frac12\cdot{-1/n\choose+1/n}^{-1}$$ Motivation: In my previous question, thanks to Will Jagy’s simple but brilliant answer, I was able to express the area of the superellipse $x^n+y^n=r^n$, for odd values of $n=2k+1$, with $k\in\mathbb N^*$, as $A_n=r^2\displaystyle\cdot{2/n\choose1/n}^{-1}+r^2\cdot{-1/n\choose+1/n}^{-1}$, where the first term, […]

Prove that $\Gamma(p)\times \Gamma(1-p)=\frac{\pi}{\sin (p\pi)},\: \forall p \in (0,\: 1)$

Prove that $$\Gamma(p)\times \Gamma(1-p)=\frac{\pi}{\sin (p\pi)},\: \forall p \in (0,\: 1)$$ With $$\Gamma (p)=\int_{0}^{\infty} x^{p-1} e^{-x}dx$$ My tried: We have $$B(p, q)=\int_{0}^{1} x^{p-1} (1-x)^{q-1}dx=\frac{\Gamma(p)\times \Gamma(q)}{\Gamma(p+q)}$$ Hence $$B(p, 1-p)=\frac{\Gamma(p)\times \Gamma(1-p)}{\Gamma(1)}=\Gamma(p)\times \Gamma(1-p)=\int_{0}^{1} x^{p-1} (1-x)^{-p}dx$$ But, come here I don’t know how :((

Mathematical meaning of certain integrals in physics

While studying on texts of physics I notice that differentiation under the integral sign is usually introduced without any comment on the conditions permitting to do so. In that case, I take care of thinking about what the author is assuming and the usual assumption made in physics that all the functions are of class […]

Calculation of $\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x} dx\;,$

Calculate the definite integral $$ I=\int_0^{\pi} \frac{\sin^2 x}{a^2+b^2-2ab \cos x}\;dx $$ given that $a>b>0$ My Attempt: If we replace $x$ by $C$, then $$ I = \int_{0}^{\pi}\frac{\sin^2 C}{a^2+b^2-2ab\cos C}\;dC $$ Now we can use the Cosine Formula ($A+B+C=\pi$). Applying the formula gives $$ \begin{align} \cos C &= \frac{a^2+b^2-c^2}{2ab}\\ a^2+b^2-2ab\cos C &= c^2 \end{align} $$ From […]

Understanding the solution of $\int\left(1-x^{p}\right)^{\frac{n-1}{p}}\log\left(1-x^{p}\right)dx$

I would like to solve the integral $$\int\left(1-x^{p}\right)^{\frac{n-1}{p}}\log\left(1-x^{p}\right)dx.$$ My problem is that I arrived at a solution via wolfram alpha, but I would like to understand how one would arrive there by hand. What I did is to use the substitution $x=\left(1-\exp(z)\right)^{\frac{1}{p}}$. This yields $$-\frac{1}{p}\int\left(1-\exp(z)\right)^{\frac{1}{p}-1}\exp\left(\frac{n-1+p}{p}z\right)zdz$$ for which wolfram alpha yields the integral \begin{align} &\frac{\exp\left(z\frac{(n+p-1)}{p}\right)}{\left(n+p-1\right)^{2}}p\,_{3}F_{2}\left(1-\frac{1}{p},\frac{n}{p}-\frac{1}{p}+1,\frac{n}{p}-\frac{1}{p}+1;\frac{n}{p}-\frac{1}{p}+2,\frac{n}{p}-\frac{1}{p}+2;\exp(z)\right)\\&-z\frac{\exp\left(z\frac{(n+p-1)}{p}\right)}{\left(n+p-1\right)}\,_{2}F_{1}\left(\frac{p-1}{p},\frac{n+p-1}{p};\frac{n+2p-1}{p};\exp z\right). […]

Why do we need to learn integration techniques?

After a lifetime of approaching math the wrong way, I took two college math courses this quarter with a newfound zest for math. These classes are integral calc and multivariable calc. Integral calc started out okay, learning about Riemann sums and the Fundamental Theorem of calculus. But instead of spending a great deal of time […]

Integral $\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3)$

$$ I=\int_0^{\pi/2} \theta^2 \log ^4(2\cos \theta) d\theta =\frac{33\pi^7}{4480}+\frac{3\pi}{2}\zeta^2(3). $$ Note $\zeta(3)$ is given by $$ \zeta(3)=\sum_{n=1}^\infty \frac{1}{n^3}. $$ I have a previous post related to this except the logarithm power is squared and not to the 4th power. If you are interested in seeing this result go here: Integral $\int_0^\pi \theta^2 \ln^2\big(2\cos\frac{\theta}{2}\big)d \theta$.. However, I […]