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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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The derivative of an integral when 2 variables are involved

I was studying definite integrals and there is a property in my textbook: If $F (t)=\int_a^b g (x,t)dx $ , then $\frac {dF}{dt}= \int_a^b \frac {\partial g (x,t)}{\partial t}$ This makes sense to me as when we integrate, all the $x's $ are going to…
user185887
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Need help with this question.

I was trying to solve this question but got stuck. If we further solve it we get $I'(e)=0 $ which does no help to find the value of the integral. I know an alternate way to take $7$ as variable but I want to know if the process I used can help in…
Jasmine
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How to calculate $\int_0^1 \frac{1-x}{(x^2-x+1)\log(x)} dx$

How to calculate this integral $$\int_0^1 \frac{1-x}{(x^2-x+1)\log(x)}\;dx$$ In WolframAlpha, I found it equal to $$\log \left[ \frac{\sqrt{\pi}\;\Gamma\left(\frac23\right)}{\Gamma\left(\frac16\right)} \right]$$ I tried using the…
Wolfdale
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How can I approach this problem $\int_{-1}^{\frac32} |x\sin(nx)| dx$

How can I approach this problem $$\int_{-1}^{\frac32} |x\sin(nx)| dx$$ How to handle these type of integrals? Apart from the presence of $n$ as an argument of $sine$ function, how do I even check where it is positive or negative in the given…
So Lo
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Integrate $\int_0^{\pi} \frac{x}{\sin x} \log {\frac{1+\sin x} {1-\sin x}}\,\mathrm d x$

$\int_0^{\pi} \frac{x}{\sin x} \log {\frac{1+\sin x} {1-\sin x}}\,\mathrm d x$ I'm stuck on this one. Any ideas? I have tried substitutions and integration by parts. I managed to show it is equivalent to $\pi \int_0^{\pi/2} \frac{1}{\sin x} \log…
user544680
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Looking for a candidate function to solve definite integral

I'm working on a numerical simulation of population dynamics that would be greatly sped up if I could solve the following definite integral: $$\int_{0}^te^{-[x-\theta(T+\tau)]^2/2\sigma^2}d\tau $$ where $\theta(x)$ is any well behaved sigmoid…
dvasseur
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Integral word problem. Did I set this up correctly?

Here is the question: Use the method of cylindrical shells to find the volume generated by rotating the region bounded by the given curves about the specified x-axis. Here are the parameters: $$y = 4x - x^2, y = 3\text {; about }x = 1$$ So here…
Jwan622
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Definite integral word problem with trig. Check setup?

Find the volume of the solid obtained by rotating the region bounded by the given curves about the specified line. Sketch the region, the solid, and a typical disk or washer. $y = \sin x$, $y = \cos x$, $0 < x < \frac{\pi}{4}$; rotated about the…
Jwan622
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Is this a good case for horizontal integration? Find enclosed area btw 2 equations. Why or why not?

When doing horizontal integration... why do we write the function as a function of y with the x by itself? What is the high level point of that? So here is my work. Is this right? I am asked to find the area enclosed between these equations: $$ 4x +…
Jwan622
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Definite integrals proof.

I am reading this text and I don't get the proof: I don't see how the FTC2 is applied a second time. Where did the second set of equations come from? Where did the $u$ come from? And by "second time" do they mean "let's apply FTC2 to a completely…
Jwan622
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Arrangement of definite integral in increasing order

If $\displaystyle I=\int^{\frac{\pi}{2}}_{0}\cos(\cos x)dx$ and If $\displaystyle J=\int^{\frac{\pi}{2}}_{0}\sin(\cos x)dx$ and If $\displaystyle K=\int^{\frac{\pi}{2}}_{0}\cos xdx$. Then Arrangement of $I, J,K$ in increasing order is Try:…
DXT
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How can I aesthetically improve my percent light occlusion formula?

I was working with the following problem: Given a sensor along the $x$-axis from the origin to $(l,0)$, a light source at $(x_1,y_1)$, and a wall between them with one corner at $(x_2,y_2)$ and the rest extending away at the angle $\theta$ for a…
Aaron Quitta
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Double integral with Gaussian like form

I am trying to calculate the following integral: $$\int_{0}^{\infty}\int_{-\infty}^{0} e^{\sigma y}e^{c(x+y)-Tc^2/2}\frac{2}{\sqrt{2\pi T^{3}}}e^{-\frac{(x-y)^2}{2T}}dydx,$$ where $c =\frac{r-\sigma^{2}/2}{\sigma}$ and $T,\sigma,r>0$. I have tried…
Patrick
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Using definite integration to find the area enclosed by a curve

Say you have the curve $y=x^3-5x^2+6x$ with roots 0, 2 and 3. Integrating $y$ I got the equation $\frac14x^4-\frac53x^3+3x^2+C$. I found the value of $C$ by subbing a point form the curve such as $(0,2)$ and found the value for $C$ to be $-\frac83$,…
Benny
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Finding $a$ for which definite Integral is minimum

If $\displaystyle f(a)=\int^{\infty}_{0}\frac{x^a}{2x^6+4x^5+3x^4+5x^3+3x^2+4x+2}$is minimum.Then real value of $a$ is Try: $$f(a)=\int^{\infty}_{0}\frac{x^{a-3}}{2(x^6+x^{-6})+4(x^2+x^{-2})+3(x+x^{-1})+5}dx$$ given $x>0$. So using A.M$\geq$ G.M,…
DXT
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