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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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If $u^2 = 2x-1$, why does $du^2 = dx$?

So the problem i encountered was: $$\int_1^5\frac{x}{\sqrt {2x-1}}dx$$ the first step in the given solution was: $$u^2 = 2x-1 \Rightarrow 2du^2 = 2dx \Rightarrow udu ~\text{(or} ~ du^2\text{)} = dx,$$ which confused me, as the derivative of $u^2$…
Axe
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Definite integral problem.....

And here I got stuck. What should I do next?
Arishta
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A definite integral involving a parameter

Does this integral seem to have a nice closed form (at least for a subset of values of $a > 0$) $$ \int_{\sqrt{a}}^\infty \frac{y^2}{(y^2-a+1)^2} dy $$ Using a symbolic math software and for $a$ being some small integers, I end up getting different…
passerby51
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A definite integral involving a logarithm and trigonometric functions.

The question is to compute a definite integral below. I have expressed it as a series expansion in powers of $r$ by expanding the integrand in a series and integrating term by term. Then I input the series into Mathematica and Mathematica returned…
Przemo
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$f$ is increasing and concave up, determine definite integral

Can you help me? I understand that it has to do with areas. I can draw pictures and unsteratsnd thet the inequality holds. But how to prove it correctly?
Sofia.T
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appearance of curve $z= xy$ on intersection with surface $x^2+y^2=a^2$

What does the curve of $z=xy$ look like ? To find the area of the surfaces for the part of $z=xy$ cut off by cylinder $x^2+y^2=a^2$ what will the intersection of the two surfaces look like?
shadow kh
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How to proceed this definite integral?

$\int_{\frac{1}{2014}}^{2014}\frac{\arctan x}{x}\text{d}x$
Resorcinol
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solving Integration of trignometry

In the following integral $$\int \frac {1}{\sec x+ \mathrm {cosec} x} dx $$ My try: Multiplied and divided by $\cos x$ and Substituting $\sin x =t$. But by this got no result.
Koolman
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$\int\limits_{-\infty}^{\infty}(f(x))^{a} dx$ , $0 < a < 1$ given $\int\limits_{-\infty}^{\infty}f(x) dx$

For any $f(x)$, are there any tricks to compute $\int\limits_{-\infty}^{\infty}(f(x))^{a} dx$ , $0 < a < 1$ given a known $\int\limits_{-\infty}^{\infty}f(x) dx$ ? I thought differentiation under integral sign might lead me to something like this,…
Srini
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Can this integral involving Gaussian and sine functions be written in terms of special functions?

Recently, I encountered the following integral. Given that it is relatively of a simple form, I suspect it is written in terms of known special functions. I know for $\phi = \pi/2$, it corresponds to Craig formula. $$ I(a, \phi) = \int_{0}^{\phi} …
Sungmin
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Definite integral of exponential of nested function

I struggle to solve the following definite integral: \begin{equation} \int_{-a}^{a} e^{\frac{1}{c_0+c_1\cdot cos(x)}} \mathrm{d}x \end{equation} where $c_0\neq 0$, $c_1\neq 0$ and $a > 0$ The substitution approach to a linear expression in the…
Peter Power
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Evaluation of $\int^{1}_{0}\frac{\ln^2(1+x)}{x}dx$

Evaluation of $$\int^{1}_{0}\frac{\ln^2(1+x)}{x}dx$$ $\bf{My\; Try::}$ Let $$I = \int^{1}_{0}\frac{\ln^2(1+x)}{x}dx = \int^{1}_{0}\ln^2(1+x)\cdot\frac{1}{x}dx$$ Using By parts, We get $$I = \left[\ln^2(1+x)\cdot…
juantheron
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Proof of existance for a specific integral

Probably just dont see it, was never good with prooving existance. Prove that this integral exists $$\int_0^4dx \int_{\sqrt x}^{min[2,2\sqrt x]} \frac{dy}{\sqrt {x+y^2}}$$ Then calcualte. I would start with $$\int_0^4dx \int_{\sqrt x}^{min[2,2\sqrt…
Katpton Liamfuppinshire
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Find volume by using triple integration

I have trouble setting up a triple integral to find volume bound by equations, such as: I'm not sure how to figure how to find the boundaries for the $z$-axis. How would I go about setting up the triple integral ? Find the volume bounded by $z^2 = …
Johhnnyy Elvis
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Is there a closed form for the integral $\int_u^{\infty}(x+a)^ve^{-bx}dx$

I'm looking for the close form of the integral $\int_u^{\infty}(x+a)^ve^{-bx}dx$, where $u,a,b$ are positive real numbers (b could be integer in special case), $v$ is a complex number. Are there anyone aware of it? Thanks in advance!
kawofengche
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