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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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How to compute the integral $\int_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}} \ dx$

I want to compute this integral $$\displaystyle\int_{0}^{\infty} \frac{x^{1/3}}{1+x^{2}} \ dx$$ What I did was the following. I substituted $x=t^{6}$, so that my $dx= 6t^{5} \ dt$ and so the integral changes to $$\int_{0}^{\infty}…
Seshadri
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Find all continuous function $f(x)$ such that $[f(x)]^2 = \int_{0}^{x} \frac{t f(t)}{1+t^2}dt$

Find all continuous function $f(x)$ defined real fro all real $x$ such that: $$[f(x)]^2 = \int_{0}^{x} \frac{t f(t)}{1+t^2}dt$$ I am really stuck. Don't we have to differentiate both sides with respect to $t$? $$2[f(x) f'(x)]\frac{dx}{dt} =…
A. Radek Martinez
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Water level changing with definite integrals.

The water level of a certain body of water is changing at a rate of $W(t)=\dfrac12\cos\left(3-\dfrac t2\right)$ inches per hour, where $t$ represents hours since $12$ a.m. (A.) Write the integral that represents the number of inches that the water…
Ayanoria
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Help in proving that $\int_{0}^{\infty} \frac{dx}{1+x^n}=\int_{0}^{1} \frac{dx}{(1-x^n)^{1/n}},n>1$

I wanted to prove that $$\int_{0}^{\infty} \frac{dx}{1+x^n}=\int_{0}^{1} \frac{dx}{(1-x^n)^{1/n}}, n>1.$$ I converted them to Gamma functions but I could not prove it. Please help me.
Dharmendra Singh
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How to solve $x^2 + \ln(x) = 0$

I was just investigating $y = f(x) = e^{-x^2}$ and then went ahead to plot $x=f(y), y=-f(x), and x=-f(y)$, and what I got was interest rounded square shape, and I think we can calculate this area using integration. However to get the bounds I must…
Mrigank Vallabh
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Is the following true $\frac{\int_0^{4\pi}e^x(\sin^6 ax + \cos^4 ax)\,dx}{\int_0^{\pi}e^x(\sin^6 ax + \cos^4 ax)\,dx}=\frac{e^{4\pi}-1}{e^{\pi}-1}$?

If $f(x)=e^x(\sin^6 ax + \cos^4 ax)$ where $a\in\mathbb Z$. Let $S_1$ be the area of the region bounded by $y = f (x)$, with x-axis and between the ordinates $x=0$, $x=4\pi$ , let $S_2$ be the area of the region bounded by $y =f(x)$, with x-axis and…
user797330
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Definite integral $\int_0^\infty \frac{\sin^4(7x)-\sin^4(5x)}{x} \ \mathrm d x$

Find the definite integral of: $$\int_0^\infty \frac{\sin^4(7x)-\sin^4(5x)}{x} \ \mathrm d x$$ This question is from a Belarusian mathematical olympiad. This is from the topic of definite integrals, but I can't think of anything. I tried using the…
Obsessed with integrals
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If $\int_0^{10} g(x)\,dx = 8$, what is $\int_0^{10} [g(x)+2]\,dx$?

Consider the function $g$ that is continuous on the interval $[−10, 10]$ and that $\int_0^{10}g(x)dx=8$. What is $\int_0^{10}[g(x)+2]dx$ equal to? So I tried just substituting the first function to get $8+2$ to equal $10$, but that's wrong. I really…
Ayanoria
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Rotational Volume of $y^2 = \frac{x^3}{2a-x}$ around asymptote $x = 2a$

My approach is to use a shell method over the range $[0, 2a]$. One cylinder will be $C = 2\pi xy \ dx$. Let us only work with the positive quadrant and multiply by two for ease: https://www.desmos.com/calculator/pvuquz8orz then the volume is: $$V =…
A. Radek Martinez
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Integral wrt floor(x)

What is the definite integral of $f(x)=x^2+1$ with respect to the differential of $\lfloor x\rfloor$ i.e ($d\lfloor x\rfloor$) from $0$ to $2$? I tried to multiply and divide dx by then $d\lfloor x\rfloor/dx = 0$. How do I approach it?
Harrison Wells
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Proof that the upper and lower Darboux sums approach the same limit.

While finding the area under the curve we break the curve into infinite numbers of strips and then find the area of individual strips. First we take circumscribed strips and then inscribed strips. They lead to an overestimation and underestimation…
ronak jain
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Is $\int_{0}^{\infty} e^{-(a^2x^2+\frac{b^2}{x^2})}dx=\int_{0}^{\infty}e^{-(b^{2}{X}^2+\frac{a^2}{X^2})}dX$ for arbitrary $a,b$ and fixed range?

The problem says, If $\int_{0}^{\infty} \mathbb{e^{-(a^2x^2+\frac{b^2}{x^2})dx}=\frac{\sqrt{\pi}}{2a}.e^{-2ab}} \longrightarrow(i)$, then prove that…
vbm
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Proving an integral relation

for the following Question, i had to prove this : that for every $$-1\le y \le 1 \\ \arcsin(y) + \arccos(y) = \frac{\pi}{2}$$ NOTE: this I've shown this using basic trigonometric id's and (probably) somehow use this to prove the following :…
Simba
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Darboux integrable periodic function is integrable on any closed and bounded interval

How do I show that a Darboux integrable periodic function, with period $T$, is integrable on any closed and bounded interval? Intuitively it is so clear to me, but how one can provide a proof? Thank you for any help!
17SI.34SA
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Can definite integrals be decomposed this way?

Suppose a function is defined as $I(x) = \int_{0}^{x}g(u)du.$ Does there exist constant bounds of integration $I(c)$ such that $I(x) - I(c) = \int_{0}^{x-a} g(u)du.$ Or in other words, is it possible to lag the boundary of $I(x)$ by some constant…
CheeseMongoose
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