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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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Change value of integrand at finite number of points

What happens to a riemann integral value if I change a finite number of points of the integrand ? I've read it stays the same but don't know the proof.
SotirisD
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Applying domination rule of integrals

For the inequality $\sin x \le x$ which holds for $x \ge 0$, Find the upper bound for the value of $\int_0^1 \sin x dx$, express it in a simplified fraction or an integer. Domination rule states that $\int_a^b f(x) dx \ge \int_a^b g(x) dx$ if…
user307640
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How do I apply the Riemann sum with equal-width sub-intervals?

$$\int^b_af(x)dx=\lim_{n\rightarrow\infty} \sum^n_{k=1}f\left(a+k\cdot\frac{b-a}{n}\right)\left(\frac{b-a}{n}\right)$$ I am trying to apply this to $$\int_0^4 (4x+2) dx$$ In this scenario, what is $n$? is $n=4$?
user307640
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What can be the specific reason for this question?

I have tried this question by drawing the graphs of the above two functions and found that both are giving asymptote at $Y=0$. But surprisingly integration of $\ln{(x)}$ from $0$ to $1$ is defined but integration of $\text{pow}(x,\ln{(x)})$ is…
DEB SANKAR ROY
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is $dx$ (at least in integrals) a positive real number?

Given the Riemann integral: $\int_a^b f(x)dx$ By definition: $dx = lim \frac{b-a}n$ as $n\rightarrow+\infty$. So, is it $dx$ a real number and a positive one as it seems apparent by defintion? There seems to be a lot of confusion in literature about…
Giack_89
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Definite integral - working rules

I am trying to solve the definite integral given below. $ I = \int_{0}^\pi \frac{1}{cos^2x + 4sin^2x} dx $ Shown below is the plot of the integrand function (as shown on wolfram alpha). It doesn't show any discontinuity in$[0,\pi]$ interval. The…
Chintan S
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Help with splitting the interval of an integral.

I have the integral $$\int_{t=1}^{u}\min(1,t^{\,n-1}(u-1)^{n-1})\,dt .$$Can somebody kindly help with splitting the interval of integration $(1,u)$ so as to get rid of the $min$ function? Since $1 \leq t \leq u$, $$ 1\leq t^{n-1}(u-1)^{n-1}…
AgnostMystic
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$\int_0^{\infty} \frac{\sin x}{x^p} dx$

Evaluate$$\int_0^{\infty} \frac{\sin x}{x^p} dx$$ I tried to use same trick as we do to evaluate ${sinx\over x}$ . But i couldn't find a suitable partial function to differentiate and then integrate . Also tried writing it as $\Im[ \int_0^{\infty}…
RKK
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Evaluating a definite integral, Conceptual doubt in changing limits

Evaluate $$I=\int_0^{\infty}\frac{1}{(1+x^9)(1+x^2)}dx$$ I substituted $x=\tan y$ which gives $dx=\sec^2ydy$. Now I am having problem in determining the limits of new integral formed after substitution because $\tan y=0$ and $\lim_{x\to y}\tan…
Lalit Tolani
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A definite integral problem related to sliding down frictionless QUARTER CIRCLE

SOLVE FOR $\int_0^{R} \frac{1}{\sqrt{y(R^2 - y^2)}} dy$ THEORY BELOW Let there be a block of unit mass sliding down a circular curve of radius $R$ such that it started with $0$ initial speed from top most point as shown. I attempted to find the…
Anshuman Dwivedi
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Evaluating $\iint_S \sqrt{1+4x^2}\, dS$

I have to compute $\iint_S \sqrt{1+4x^2} dS$ and I'm looking a way to do it. The graph seems to be: the surface $S$ is given by ($x=0, x=2, y=0, y=3$ and $z=x^2$) Can anyone help me set up the integral to compute it?
Valent
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How can I write $x^2+y^2=4x$ in polar coordinates limits?

How can I write $x^2+y^2=4x$ in polar coordinates limits? Suppose $D$ is the region $x^2+y^2=4x$. After some computations and transformations using green theorem, I become to this integral : $\iint_{D} x^2+y^2 dx\ dy$ and I want to write it as…
Valent
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Trying to reduce expression

I am working through a derivation in Nielsen and Chuang's 2016 text book Quantum Computation and Quantum Information, and I have not been able to reproduce a result they get for Equation 5.34. While the text is not a math text, this question is.…
Anne
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Solving an integral with ln in the numerator and denominator

$$\int_4^8 \frac {\ln(9-x)}{\ln(9-x)+\ln(x-3)}\,dx$$ So far I've tried using u substitution with $u = 12-x$, which is what my teacher recommended, and put that equation in terms of $x = 12-u$ and plugged that in, but I got stuck there. Any and all…
user978757
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Definite integral $\arctan$ over $x$

$$\int_0^{\infty}\frac{\arctan(2x)-\arctan(x)}{x }dx$$ I checked that the integral converges. Next, the only thing that seems to appear to me to do is arctan sum formula $$\int_0^{\infty}\frac{\arctan({x\over{1+2x^2}})}{x }dx$$ but then it's dead…
RKK
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