Linked Questions

I'm looking for a closed-form of the following integral problem. $$I = \int_0^{\pi/2} \arctan(x)\cot(x)\,dx.$$ The numerical approximation of $I$ is $$I \approx 0....

I keep trying to solve this but I end up needing to do integration by parts like 3 or 4 times. My only question is, is that going to be the only way to do this? If so it will literally take me hours. ...

This is from an integral problems book, it is a part of the solution of a larger problem. $$\int_0^1\frac1{(x^2+4)^2}~dx=\frac18\Biggl( \Bigl(\frac x{x^2+4} \Bigr) \bigg\vert _0^1~+~\int_0^1\frac{dx}{...

I know how to integrate, for example $x^2\sin x$ via integration by parts. But how would one approach $x^n\sin x$, where $n$ is an arbitrary natural number? Do I have have to use integration by parts $...

The integral $$\int\frac{1}{(x^2+a^2)^m}dx$$ can be expressed by a recursive formula of $$\frac{1}{2a^2(m-1)}\frac{x}{(x^2+a^2)^{m-1}} + \frac{2m-3}{2a^2(m-1)}\int\frac{dx}{(x^2+a^2)^{m-1}}$$ I do not ...

I need to evaluate $$ \int_0^1 \dfrac 1{(x^2+1)^2}. $$ I was absent for this lecture and I'm sort of lost on where to go. I set $x = \tan (\theta)$ and $dx = \sec^2(\theta)\, d\theta$. And this ...

I am integrating $$\pi\int\left(\cfrac{1}{x^2+4}\right)^2dx$$ I understand I need to integrate by parts. But why is it that $$u=\cfrac{1}{x^2+4}$$ and not $$u=\left(\cfrac{1}{x^2+4}\right)^2$$ How ...

How can I integrate: $$\int \frac{1}{(t^2+k^2)^m}\, dt$$ without trygonometric substituition? where $t= (x+(p/2))$ and $k= (1-(p^2/4))$ coming from an equation with complex roots: $x^2 + px + q.$ ...

Compute $$\int_{-\pi/2}^{\pi/2} \frac{x^2\cos(x)}{1+e^x}$$ The first step itself given in the solution is changing $e^x$ to $1/e^x$ . Now as first step is making no sense to me so I didn't post the ...

How to calculate the following integral if $\varepsilon \in (0,1)$: $$\int \limits_{0}^{\pi}\frac{d\varphi}{(1+\varepsilon\cos \varphi)^2}$$

I would like to know if there is an algebraic way to evaluate $$\int \frac{\text{d}x}{x^2(x^2+1)^2}$$ Without using partial fraction decomposition and without substitution. My attempt is something ...

I know that $\displaystyle\int \dfrac1{ax^2+bx+c}\,\mathrm dx$ is easily solvable using completing the square, but my question is how would would find $$\displaystyle\int\frac{1}{\left(ax^2+bx+c\...

So I was trying to use partial fraction decomposition on this problem, and I realized that it didn't work, as it is already in partial fraction decomposition form. $$\int{\frac{3x+4}{(x^2+5)^2}dx}$$ ...

I need to calculate: $$\int x^2 \cdot \cos{x} \cdot dx $$ My step by step solution: Using the formula: $$\int udv=uv-\int vdu$$ $\color{gray}{\boxed{\color{black}{\int x^2 ⋅ \cos{x}⋅ dx=x^2 \sin{x}-...

Recently I've been looking around for integral tricks here, here, and here (just to name a few). I came across the post on brilliant's website here (move the slider/ scroll down to the very bottom to ...