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Questions tagged [definite-integrals]

Questions about the evaluation of specific definite integrals.

A definite integral is defined as the area under a function from $a$ to $b$. Definite integrals sometimes involve calculating the indefinite integral, which is a function giving the area from $0$ to any $x$. However, definite integrals are most often separate from indefinite integrals in that the indefinite integral may not exist on its own. This is usually in the case of piece wise functions that are split along certain key points or integrals involving asymptotes.

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Finding the definite integral $\int_{0}^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx$

$$\int_0^{2\pi} \frac{e^{|\sin x|}\cos(x)}{1+e^{\tan x}} \, dx$$ My try: $$I=\int_0^\pi \frac{e^{\sin x}\cos(x)}{1+e^{\tan x}} dx+\int_\pi^{2\pi} \frac{e^{-\sin x}\cos(x)}{1+e^{\tan x}} dx$$ also $$I=-\int_0^\pi \frac{e^{\sin x}\cos(x)}{1+e^{-\tan…
user220382
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an integral problem

$$ \int_{-\infty}^{\infty} [c_1 + c_2 (x-c_3)^2 + (x - c_4)]^{-c_5} \, dx $$ with $c_1, c_2, c_3, c_4, c_5$ known real constant. Can you help me to solve this integral?
user7064
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Definite integral question

Let $ f(x)$ be a quadratic equation with $f'(3)=3$. If $I=\int_{0}^{\frac{\pi}{3}}t \times \tan(t)dt $ and the value of integral$\int_{3-\pi}^{{3+\pi}}f(x) \times \tan(\frac{x-3}{3})dx $ is equal to $kI$.Then find k. My attempt: Put…
Vinod Kumar Punia
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$\int_1^\infty \operatorname{sech} x \cdot \ln x \ dx$

The integral cant be expressed in "standard mathematical functions" -Wolfram I'm asked to determine if its convergent or divergent (I can do that via comparison theorem) and its convergent (and wolfram agrees) but them I'm asked to evaluate because…
Nawar Ismail
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Integrating fractional part function

If $f(x) = \min(\{x\},\{-x\})$, then find $$\displaystyle\int_{-100}^{100} f(x)\, {\rm d}x,$$ where $\{x\}$ denotes fractional part of $x$.
Kushashwa Ravi Shrimali
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Definite integral application

So, the task is to calculate the area of a shape in xOy plane bounded by functions: $y = x\sqrt{4x-x^2}$ and $y = \sqrt{4x-x^2}$ Could you please explain how I can solve this? How can I find the intersection points of these functions and how do I…
A6SE
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Integrate $ \sin x /(1 + A \sin x)$ over the range $0$,$2 \pi$ for $A=0.2$

Wolfram Alpha indicates the following solution form:- $$ \int_0^{2\pi} \frac{\sin x}{1 + A \sin x} dx = (1/A)\left( x - \frac{2 \tan^{-1} \left( \frac{A + \tan{(x/2)}}{\sqrt{(1-A^2)}}\right)}{\sqrt{(1-A^2)}} + constant\right)^{2\pi}_0 $$ My…
steveOw
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Integrals in 1-dimention

Probably most of you know a short article by Terence Tao on differential forms (http://www.math.ucla.edu/~tao/preprints/forms.pdf). He talks about three integration concepts, two of which are the definite integrals. The integration of functions on…
rk85
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How to interpret a definite integral outside the domain of the function?

Considering the geometric interpretation of the definite integral (finding the area under a curve) what should I do if the interval of integration is not contained in the function's domain? For instance $$ \text{Calculate} \int_{-2}^4 f(x)dx \quad…
Карпатський
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Volume of Hyperboloid

What would be the easiest way to calculate the volume of Hyperboloid with inequation from the picture. I don't know how to approach this problem.
user18960
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Estimate from below $\int_0^\pi e^{-t}\cos nt dt$ without calculate it.

Estimate from below the following integral $$\int_0^\pi e^{-t}\cos nt dt$$ without calculate it. Here $n\in\mathbb N$. Any suggestions please?
Mark
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How to solve problems of this type?

I would like to know a nice approach for problems of this type $I_n = \int_{-\pi}^{\pi} \frac{\sin (nx)}{\sin x} dx$ Then find $I_n$ My Approach:- Trying to write $I_n$ as a difference of two integrals, and then summing, in a telescopic fashion.…
Bayern ROX
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How to solve $\int_{-1}^{1} (x^{4/3} + 4x^{1/3}) dx$?

I started by integrating it this way: $$(\frac{3}{7} x^{7/3} + \frac{16}{4}x^{4/3})$$ What is wrong with it?
juliano.net
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Is there a name for the relation $\int^b_a f(x)\ dx=\int_a^c f(x)\ dx+\int_c^b f(x)\ dx$?

The question is contained in its title. French mathematicians would call such a relationship "Chasles' relationship". This comes for the similarity with the formula $\vec{AB}=\vec{AC}+\vec{CB}$ for vectors that is also called "Chasles'…
Taladris
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Find the proper point on astroid for conditions to be met, using definite-integral for finding length.

I was given the following task: We have an astroid $x=acos^3(t), y=asin^3(t)$ and two points on it $A(a,0), B(0,a)$. We need to find point $M$, which belongs to the arc $AB$, and the length of arc $AM$ should be equal to $\frac{1}{4}$ length of arc…
blitzar787
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