0
votes
2answers
79 views

Integral $\int\sqrt{\sin2x}\operatorname d\!x$

I tried all substitutions but failed. I need assistance to evaluate that indefinite integral. $\int\sqrt{\sin2x}\operatorname d\!x$
0
votes
1answer
27 views

Find Indefinite of root function

I don't know how to find this strange integral $\int{\sqrt{\dfrac{x-4}{x+2}}\dfrac{dx}{x+2}}$ Please help me solve this problem
1
vote
3answers
70 views

If $f:[0,1]\rightarrow\mathbb{R}$ is a function such that $f=0$ over a dense set in $[0,1]$ so $\int_0^1 f=0$?

If $f:[0,1]\rightarrow\mathbb{R}$ is a Riemann-Integrable function such that $f=0$ over a dense set in $[0,1]$ so $\int_0^1 f=0$? I'm thinking about it but without progress. Would someone ...
1
vote
2answers
228 views

The unbeatable $\int e^{1/\cos(x)} dx$ integral

Is there any way to express this in non-elementary functions? $$ \int e^{1/\cos(x)} dx$$ And/or to calculate this definite integrals? $$ \int_{-\pi/2}^{\pi/2} e^{1/\cos(x)} dx$$ $$ ...
1
vote
3answers
103 views

Find the indefinite integral $\int\frac{(x+1)e^x}{x(1+xe^x)}dx$

Find the indefinite integral $$\int\frac{(x+1)e^x}{x(1+xe^x)}dx$$ I feel like this function does not have an anti-derivative in the form of elementary functions.
7
votes
2answers
200 views

When we can change $\int$ and $\sum$ for indefinite integral?

I know, for example, that if the series $\displaystyle\sum_{n=1}^{\infty}f_n(x)$ consisting of integrable functions on a closed interval $[a, b] \subset \mathbb{R}$ converges uniformly on that closed ...
1
vote
1answer
162 views

Let $f:[0,1]→\mathbb{R} $with $f′(x) $continuous. It is known that $\int_{0}^{1} f(x)dx=0$.

Let $f:[0,1]→\mathbb{R}$ with $f'(x)$ continuous. It is known that $∫_0^1 f(x) dx=0$. Prove that $∀α∈[0,1]$, $$|\int_{0}^{\alpha} f(x) dx |≤ \frac{1}{8} sup_{(0≤x≤1)}|f'(x) |$$ My answer so far ...
9
votes
2answers
317 views

Evaluate $\int\sin(\sin x)~dx$

I was skimming the virtual pages here and noticed a limit that made me wonder the following question: is there any nice way to evaluate the indefinite integral below? $$\int\sin(\sin x)~dx$$ Perhaps ...
0
votes
1answer
148 views

Derivative of the indefinite integral and Lebesgue point

Give an example where the derivative of the indefinite integral exists at point that are not Lebesgue points.
5
votes
2answers
187 views

Evaluating $\int \cos(x) \sqrt{\sin(2 x)} dx$

Evaluate the following indefinite integral: $$\int \cos(x) \sqrt{\sin(2 x)} dx$$ Only hint I have is from W|A that expresses the integral in terms of a hypergeometric function and it looks ...
5
votes
2answers
293 views

Evaluating: $\int \frac{t}{\cos{t}} dt$

How would you evaluate the following indefinite integral? In fact, I did evaluate $\int \frac{\cos{t}}{t} dt$ by parametric integration and then I thought of this variant. $$\int \frac{t}{\cos{t}} ...