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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.

20559 questions
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Sum of integration over discontinuity

Let $f(x): \mathbb{R} \to \mathbb{R^+} $ be the function: $$ f(x)=\sqrt{17 x^2+x^4}, $$ whose plot is: By integrating the function, I got: $$ F(x)=\frac{\left(x^2 \left(x^2+17\right)\right)^{3/2}}{3 x^3}+C, $$ whose plot is: When I evaluate…
Onizuka
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Cartesian to Polar Coordinates

If I want to change the following integral from Cartesians to Polars: $$\int_{-\infty}^\infty\int_{-\infty}^\infty (x-a)^2+(y-b)^2\,\,dx\,dy$$ in a way such that we are centered at $(a,b)$, so $(x-a)^2+(y-b)^2=r^2$, Is the polar form simply…
Georgie
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Definite integral from $-\infty$ to zero of an exponential function

I'm trying to compute a definite integral from $-\infty$ to zero of an equation of the form $$x^{2n+1} e^{-x^2 /a^2}$$ I have been able to find from a table in my book that the definite integral of the form with which I'm working, from zero to…
Morg Man
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Evaluating an integral that gives an undetermined infinity form

I'm solving a definite integral where one of the borne is infinity. When I try to evaluate the borne at infinity, I'm getting stuck, because I'm getting the undetermined infinity form $ 0 \cdot \infty $. Here is the integral I'm trying to evaluate…
HoLyVieR
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Integration Proofs

Prove Show that for $x\geq 1$ $\displaystyle\left|\int_1^x \frac {\sin(t)}{t} dt\right|$ $ \leq \ln(x) $ Attempt: $\displaystyle\left|\int_1^x \frac {\sin(t)}{t} dt\right|$ $ $ $ \leq $ $\displaystyle\int_1^x$ $\left| \dfrac {\sin(t)}{t}\right|…
smaMATH
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How to calculate $\int_{b}^{c} (c-s)^{-3/4} (s-b)^{-3/4}ds$

How to calculate the integral $$ \int_b^c \! (c-s)^{-3/4} (s-b)^{-3/4}\,\mathrm{d}s $$ Thanks.
Richard
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How to evaluate the following definite integral?

\begin{equation} \int_0^{2 \pi}\int_0^{2 \pi} \frac{\cos\left(u\right) + \cos\left(v\right)-\cos\left(u\right)\cos\left(v\right)}{\,\sqrt{\,\left\{\vphantom{\LARGE A}2 \cos\left(u\right)\left[\cos\left(v\right)-1\right]-2\cos\left(v\right)+3…
Upax
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Integrate $\int_a^b \ln(x-a)\ln(x-c)\,\mathrm{d}x$, where $c < a < b$?

Is there an analytical expression for the following integral? $$\int_a^b \ln(x-a)\ln(x-c)\,\mathrm{d}x,$$ where $c < a < b$ are all real numbers. I don't really need to know how this integral is done (though if you post that too it would be nice). I…
a06e
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Evaluate a definite integral

Evaluate: $$\int_{0}^{1} x^2\sqrt{(kx^2-1)(kx^2-1+t)}\, dx$$ where k and t are constants. Can you help me?
user154564
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Application of Definite Integral

I been solving this problem for an hour, but I cant get the correct answer. Please help me with this.
Hermoso
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$\int_{B}^{C}\left(x-A\right)^{\alpha}\left(x-B\right)^{\beta}\left(C-x\right)^{\gamma}\left(D-x\right)^{\delta}\mathrm{d}x$, with $A\le B\le C\le D$

Let $\alpha,\beta,\gamma,\delta > -1$, and $A\le B\le C\le D$. Is there an analytical expression, or an efficient numerical technique, to estimate the following…
a06e
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finding integrals by parts

$$\int^{1}_{0} x^2(x+1)^5 dx$$ I would appreciate if you could go through the process step by step. I have been working on this problem for quite awhile and cannot seem to figure it out. after finding the $u, du, dv, v$ my equation…
user149420
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Evaluate $\int_{-\pi}^{\pi} {2x(1+\sin x)\over1+\cos^2x} \,\, \mathrm{d}x$

This is my question, $$\int_{-\pi}^{\pi} {2x(1+\sin x)\over1+\cos^2x} \,\, \mathrm{d}x$$ (1) $\pi^2\over4$ (2) $\pi^2$ (3) zero (4) $\pi\over2$ I first broke the function into parts: $\int_{-\pi}^{\pi} {2x\over1+\cos^2x}$+ $\int_{-\pi}^{\pi}…
Yash Lekhwani
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Definite integrals-with modulus function

The greatest value of the function $$f(x) =\int_1^x | t |dt$$ on the interval$[{-1\over2},{1\over2}]$is given by: (1)$\frac 38$ (2)${-1\over2}$ (3)${-3\over 8}$ (4)${2\over 5}$ I have quite a problem dealing with this question.Should I…
Yash Lekhwani
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How to evaluate the definite integral $\int_0^x t^{n-1}e^{-(a+bt)}dt$

How to evaluate the following definite integral $\int_0^x t^{n-1}e^{-(a+bt)}dt$, where $n\in\mathbb{N}$ and $a,b>0$.
Litun
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