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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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For a continuous function $f$, find the value of $\int_{0}^{\infty}[ f(x^n+x^{-n}) \frac{\log x}{x} +\frac{1}{1+x^2}]dx$

The expression simplifies down to $$ \int_{0}^{\infty}\left[% \operatorname{f}\left(\,{x^{n} + x^{-n}}\,\right)\,\frac{\log\left(x\right)}{x} \right]\mathrm{d}x +\frac{\pi}{2} $$ I have a feeling that th first integral would end up being zero, but I…
Aditya
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Setting up limits of integration in this double integral

$$\int \int xy dA$$ where D is the region bounded by $y = x - 1$ and parabola $y^{2} = 2x + 6$ Quick question, what does the integrand represent? I see that 5,4 is an intersection point, but 5 * 4 = 20 and I'm not sure what the integrand is even…
Jwan622
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Sketch and evaluate integral.

I know how to solve the integral but I"m a bit confused about what to sketch. Can someone help me out? The integral I need to solve is this: $$\int_0^1 \int_{x^{2}}^x (1 - 2xy)dydx$$ $$\int_0^1 \left[(y - xy^{2}) \right]_{x^{2}}^xdydx$$ $$\int_0^1…
Jwan622
  • 5,704
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How do you prove the following integral result

How do you prove the result? $$\int_{0}^{\infty} \frac{\sin(\pi x)}{x(1-x^2)}dx = \pi $$
Callie12
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How to solve the integral $\int_{-\infty}^{\infty} i (k^2 + a)^{1/2} e^{ikx} dk$

How to solve the following integral? $$\int_{-\infty}^{\infty} i (k^2 + a)^{1/2} e^{ikx} dk$$ and some literature on this, thanks.
Ricardo
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Closed form for integral of $\lfloor x \rfloor$?

Is there a closed form for $$\int_{n}^{n+1} \frac{\lfloor x \rfloor}{x}dx$$ I found ln$(2)$ as my answer for n=1 from my calculator, but not sure how to find the general solution.
user817934
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Need to find positive continuous function g(x) that satisfies 3 equations involving definite integrals

$\int_0^1 g(x) dx = 1$ $\int_0^1 xg(x) dx = \beta $ $\int_0^1 x^2g(x) dx = (\beta)^2 $ I was assigned this homework problem and I don't know how to solve it. I let $ g(x) = 1 $ and let $ g(x) = nx^{n-1} $ , but that got me nowhere. I am stumped.
Flinn Bella
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Where to put the parameter $a$ in integral Feyman's technique?

I have to understand how to put the parameter $a$ when it comes to generalise the function under the integral sign. For istance I tried to solve this integral $$\int_{0}^{\infty}{\frac{\sin(x)}{x^2+1}}dx$$ But I didn't know how to generalise the…
us er
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Evaluation of Rational integration with upper and lower limit is infty

Evaluation of inproper integral $\displaystyle \int^{\infty}_{-\infty}\frac{1}{x^4+64}dx$ Note that you need to use triangle inequality in your calculation. What i try:: Let…
jacky
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Reason for negative value of integral where upper bound is less than lower one

Consider the integral, $$ \int_{2}^{0} x dx$$ Evaluating, we get a negative area but why is when add up rectangles starting at x=2 and go to x=0, is the area accumulated negative? I could easily prove this by a change of bounds substitution $ x=…
tryst with freedom
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Is this integral correctly simplified?

I'm working on calculating lift of an supersonic airfoil. It involves this calculation. I did a lot of calculations, realizing it might really equal this. Can someone please read this integral, and see if I simplified it correctly? Integral…
Eric. Alexander
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Definite triple integral

I need to evaluate the following triple integral $$ \int_{-n}^n \int_0^{\sqrt{n^2-x^2}} \int_0^{\sqrt{n^2-x^2 -y^2}} \; e^{-(x^2 + y^2 + z^2)^{\frac{3}{2}}}\; dz dydx $$ I converted this into the spherical coordinates. And the limits are as…
user9026
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Determine the following definite integral

The value of $\int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \frac{x_{1}+x_{2}+x_{3}-x_{4}}{x_{1}+x_{2}+x_{3}+x_{4}} d x_{1} d x_{2} d x_{3} d x_{4}$ In the given solution: $\int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \int_{1}^{2} \frac{x_{i} d…
user791682
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definite Integral with limit approches $\infty$

Evaluation of $$\lim_{n\rightarrow \infty}\int^{\infty}_{0}\bigg(1+\frac{t}{n}\bigg)^{-n}\cdot \cos\bigg(\frac{t}{n}\bigg)dt$$ What i Try: put $\displaystyle \frac{t}{n}=u.$ Then $dt=ndu$ $$I_{n}=\lim_{n\rightarrow…
jacky
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Finding the area between ${y=x^2}$ and ${y=2x+8}$

The two equations are ${y=x^2}$ and ${y=2x+8}$ I got the result ${-64}$ multiple times, but the proper answer seems to be different
Adis Veliu
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