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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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Help in evaluating integral using elementary methods

I have the following integral: $\int\limits_{0}^{\pi/2}\frac{x\sin x \cos x}{a^2 \cos^2x+ b^2\sin^2x} \ dx$, for $a,b \geq 0$, both not zero. I have tried several substitutions without any success at all. How can this be tackled? Thanks.
erdoswiles
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Value of this definite integral

$f$ is a differentiable function such that $f(f(x)) = x$ for $x ∈ [0, 1]$. And $f(0) = 1$. What is the value of $\int_0^1{(x − f(x))^{2016}}dx$ ? I have noticed that one solution of $f$ is $f(x) = 1-x$, then the integral can be easily evaluated. Can…
Archisman Panigrahi
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Mean value theorem for integrals, variant:

I am trying to prove the following: Suppose $\phi:[0,\infty) \to \mathbb{R}$ a continuous, bounded and integrable function and $\psi:[0,\infty) \to \mathbb{R}$ is a continuous, bounded above by 1, strictly decreasing, positive function. Then, there…
user385459
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Hints to solve this integration

I stumbled upon this integration $$\int_0^{\pi/4}(1+\cos2\theta)^2\,d\theta.$$ And I have no idea on how to proceed with it. It would be very helpful if someone would provide me with some hints. Thank you.
Ayan Shah
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what is the integral formulato get eulers' number

I've heard that there is some sort of formula where: $$\int_1^ef(x) dx=1$$ my question is: what is $f(x)$, and did i write the formula correctly?(i'm new here) $$e= 2.718...$$
Alexander Day
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computing an integral with sines and cosines

Hi i'm trying to do this definite integral $$∫_{0}^{π }\frac{\sin ^3xdx}{\left(R^2+Z^2-2ZR\cos x\right)^{3/2}}=∫_{0}^{π}\frac{\sin xdx}{\left(R^2+Z^2-2RZ\cos x\right)}-∫_{0}^{π}\frac{\cos ^2x\sin xdx}{\left(R^2+Z^2-2RZ\cos x\right)}$$ And then i…
Andrea Mora
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Kindly provide the stepwise solution for this integral.

$$\int_{-b}^b\int_{-a}^a\frac{1}{(x^2+y^2+h^2)^{3/2}}dxdy$$ First I put x=$\sqrt{y^2+h^2}tan\theta$ and arrive at: $$\int_{-b}^b\frac{2a}{(y^2+h^2)(y^2+h^2+a^2)^{1/2}}dy$$ Can you please tell me what to do after this?
Beck Humh
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Evaluate the integral 5

Evaluate the integral $ \iiint_{R} \frac{dxdydz}{\sqrt{(x-a)^{2}+y^{2}+z^{2}}} $ over the solid sphere $ 0 \leq x^{2}+y^{2}+z^{2} \leq 1 \ $ for a>1 . $$ $$ I know that the required integral is $ \int_{-1}^{1} \int_{-…
MAS
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How to find area of triangle in $1/x$ function?

I'm trying to Find a point $T[a, ?]$ on graph of function $f(x) = 1/x$, for which tangent line intersecting this point with line intersecting points $T;[0,0]$ form a triangle with area $S = 1$. I'm aware that area has to be calculated using…
John Smith
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How to find definite integral of following?

$$ \int_1^2 (2+3t)^{3/2} \ dt $$ I dont know how to solve this when $3/2$ is outside the brackets. If somebody could explain to me how to do this in the simplest way possible I would greatly appreciate it!
Jordan
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Proving that a function is integrable using the upper and lower sums

Let $a$ be a number: $0
user371583
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Is there any closed expression of this definite integral?

\begin{equation} \int_0^\theta \frac{1}{(1+e\cos x)^2} dx = ? \end{equation} in which $\theta(\le\pi/2)$ and $e$ are constants.
yangcs11
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Proving a limit of an integral

I don't know how to approach this problem. Proof that $\lim_{n\to \infty} (n+1)I_n = \frac 12$ where $I_n = \int_{0}^{1} \frac {x^n}{x+1}dx$
Catalin
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Calculate an integral of a function

I'm evaluating an integral and the function in the integral is another function as following. $$\int_0^{\infty} e^{(-0.028559\cdot t)} \frac{l(x+t)}{l(x)} dt$$ Now I have created the values of $l(x)$ in Excel for ages $x=74$ to $x=106$, because…
user333750
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Hints to perform this definite integration

I stumbled upon this integration $$\int_0^a \left(x-\sqrt{a^2-x^2}\right)^3\,dx.$$ And I have no idea on how to proceed with it. It would be very helpful if someone would provide me with some hints. Thank you.
Ayan Shah
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