Tagged Questions

116 views

$-1 = 0$ by integration by parts of $\tan(x)$

I had a calculus final yesterday, and in a question we had to find a primitive of $\tan(x)$ in order to solve a differential equation. A friend of mine forgot that such a primitive could easily be ...
36 views

$\int\sin^2(Cx)\,dx$ from a manual - need proof

In the book of quantum mechanics I came across an integral which was supposed to be from a manual ($C$ is a constant): \begin{align} \int\limits_{0}^d \sin^2\left( C x \right)\, d x = ...
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Can you explain me this antiderivative?

Find the antiderivative of $\displaystyle \frac{e^{\frac{x}{2}}}{e^x+2e^{\frac{x}{2}}+5}$. The book suggests a switch of variables. Let $t=e^{\frac{x}{2}}$. And so $x=2\ln(t)$. The antiderivative ...
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How to evaluate the trigonometric integral $\int \frac{1}{\cos x+\tan x }dx$

$$\int \dfrac{1}{\cos x+\tan x }dx$$ This can be converted to $$\int \dfrac{\cos x}{\sin x+\cos^2x}dx$$ But from here I get stuck. Using t substitution will get you into a mess. Are there ...
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How do I integrate this expression: $\int{2x\,dx\over x^3+x^{2/3}}$?

How do I integrate this expression: $\displaystyle\int{2x\,dx\over x^3+x^{2/3}}$?
1k views

Compute $\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$

I've got troubles in computing the below integral: $$\int \frac{\sin(x)}{\sin(x)+\cos(x)}\mathrm dx$$ I hope it can be expressed in elementary functions. I've tried simple substitution as $u=\sin(x)$ ...
191 views

A problem on indefinite integral: $\int(\cos x)^m\sin(nx)dx$

If $$I(m,n)=\int(\cos x)^m\sin(nx)dx,$$ how do I get $7I(4,3)-4I(3,2)$?
135 views

Evaluating $\int \frac{l\sin x+m\cos x}{(a\sin x+b\cos x)^2}dx$

How do I integrate this expression: $$\int \frac{l\sin x+m\cos x}{(a\sin x+b\cos x)^2}dx$$.I got this in a book.I do not know how to evaluate integrals of this type.
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$\int \frac{f(x) \bar f'(x)- f'(x)\bar f(x) + g(x)\bar g'(x) - g'(x)\bar g(x) }{f^2(x) + g^2(x)} \ dx$ over $\mathbb{C}$

Evaluating $$\int \frac{f'(x) g(x) - f(x) g'(x)}{g(x)^2} \ dx$$ should just give $\frac{f(x)}{g(x)}$. Now I have a similar quotient over $\mathbb{C}$, at least it looks similar. It's of the form ...
191 views

Integration of $\int\frac{1}{x^{4}+1}dx$.
I don't know how to integrate $\displaystyle \int\frac{1}{x^{4}+1}dx$. Do I have to use trigonometric substitution?
Undoubtedly, this question is so easy but I'd like to ask it. We know that the way in which the indefinite integrals like $\int P(x)e^{ax}dx$ and $\int P(x)\sin(bx)dx$ wherein $P(x)$ is an arbitrary ...