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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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Definite integral finding unknown

Given ∫1^2 f(u) du =-5, ∫1^2 h(u) du =4, ∫2^5 f(u) du =8. Find the value of p if ∫1^5 [f(u)-3pu] du=39 I just know few on how to solve this 39 = ∫1^5 f(u) du - ∫1^5 3pu du 39 = ∫1^2 f(u) du + ∫2^5 f(u) du - ∫1^5 pu du 39 = -5+8 -3∫1^5 pu du 36 =…
A Izalia
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Substitution in $\int \frac{x^2}{16x^2+1}\, dx$

Evaluate $$\int \frac{x^2}{16x^2+1}\,\mathrm dx.$$ Apparently we are supposed to let $x = \dfrac14u$ and that somehow gives $$\frac1{64}\int \frac{u^2}{u^2+1}\,\mathrm du.$$ I don't get how this happens and how do you come up with $x =…
Yanni Papaioannou
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Definite integral problem. I don't see a clear u-sub

I have this integral: $$ \int_1^2 x \cdot \sqrt{x-1} \, dx$$ I don't see it the clear u sub. If I take $ = x - 1$ then the derivative is 1 and that isn't substituable in the integrand.
Jwan622
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Definite integral involving nested square root.

If$y=\sqrt{(x^2+x+1)+\sqrt{(x^2+x+1)-\sqrt{(x^2+x+1)+\sqrt{(x^2+x+1)\cdots\cdots}}}}$. Then $\displaystyle \int^{3}_{2}ydx$. Try: writting equation as $$y=\sqrt{(x^2+x+1)+\sqrt{(x^2+x+1)-y}}$$ So…
DXT
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Definite Integral Problem up-to infinity

I was trying to solve simple integration problem, integration x from 0-infinite. Is there any particular answer to the question from any other methods? My try is I've shown on picture below.
Surya Bhusal
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definite integral of $1/(a^2cos^2x+b^2sin^2x)$

I am trying to find the definite integral of $\frac{1}{a^2cos^2x+b^2sin^2x}$ over $0$ to $2\pi$. I used the substitution $tan\ x=t$ but I am getting answer in terms of $tan^{-1}$, and that function over the given limits will be zero. But when I…
jnyan
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Integral Help: $\int^t_0{se^{d(t-s)} ds}$

I would like to know how to solve this integral, I have tried and I know the solution. Yet, I am not able at all to work around all the steps to get to the solution. I know this is a particular case, but I do not know how to achieve the right…
Lorenzo De Bernardi
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Integration: Why is this wrong?

Let $y=x(x-2)^2$. Solve for $\int {x}$ $dy$ from 0 to $\frac {9}{8}$ It is known that a point, (0.5,1.125) passes through the curve. This is the original question at part (iv) We have not learnt how to use substitution in integrals. This is evident…
ministic2001
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Square in polar coordinates

Find the area of the figure bounded by the lines given equations in polar coordinates r=φ and r=2 (Use integrals)
Макс Мотовилов
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Convergence of the function $f(x) = \int^x_0 \frac{1}{1+e^t}dt$

Is the function defined by : $$f(x) = \int^x_0 \frac{1}{1+e^t}dt$$ convergent ? When graphing it, it seems that it converges to approximately $0.6931$ : $log(2)$. The primitive of the function : $$f(x) = \frac{1}{1+e^x}$$ is $$F(x) =…
PearlSek
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What would the value of $\int_0^{\infty} \tan(x) \,dx$ be?

We cannot take the definite integral of $\tan(x)$ on the interval $[0, \pi/2]$, just like we can't take the definite integral of $1/x$ on $]0, 1]$; but what would happen if we took the definite integral of $\tan(x)$ between $[0, \infty[$ ? Would…
MattMatt
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When is $\int_0^{2a} f= 2 \int_0^a f$?

When (for what kind of $f$ and e.g. $a \in \mathbb{R}$) is $$\int_0^{2a} f= 2 \int_0^a f$$? With some trigonometric functions ($a=\pi$)?
mavavilj
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Impossible definite integral!

Is it possible to calculate this integral? $$ \int_{0}^{+\infty}\exp\left(-a\sqrt{b^2+x^2}\right)\sin(cx)\frac{dx}{x}$$ I'm tired of it. I almost used any method to calculate this definite integral. It seems there is no analytic solution. I need…
Jalil vafa
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Proof definite integral $\int_α^{π/2-α}f(\tan{x})dx=\frac{\pi}{2}-α $

Given: $$f(x) +f\left(\frac{1}{x}\right) =1$$ I am trying to prove that $$\int_α^{π/2-α}f(\tan{x})dx=\frac{\pi}{2}-α $$
GorillaApe
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Integral between limits 0 and pi

Please Integrate $$ \int_0^\pi \frac 1{(a^2\sin^2x + b^2 \cos^2)^2}\, dx$$ between limits 0 and pi. I have tried it multiple times but i keep falling into the tan inverse trap.
Achintya
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