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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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Integration of: $\frac{x}{\ln x}$

I would like to ask some assistance with how to integrate : $\dfrac{x}{\ln x}$ I could really use an explanation and final answer. Thank you very much.
user118972
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Integral comparison question

The question wants me to prove that $$\frac { \pi }{ 6 } <\int _{ 0 }^{ 1 }{ \frac { dx }{ \sqrt { 4-{ x }^{ 2 }{ -x }^{ 3 } } } < } \frac { \pi }{ 4\sqrt { 2 } } $$ My first attemp was to show that the integrand is on the interval $(\frac { \pi…
Hckr
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how to integrate : $\int_{x^2}^{x^3}\frac{dt}{\sqrt{1+t^4}}$

Need to compute: $$\int_{x^2}^{x^3}\frac{dt}{\sqrt{1+t^4}}$$ I tried to use partial fraction but got a messy algebra. thanks.
abi
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Interchanging sum and Riemann Integral

The question: Is there a theorem that allows you to interchange sum and integral for positive functions sequences, but the theorem and it's proof only involves riemann or riemann stieljes integrals. Or if Fubini and Tonelli theorems have a "Riemann"…
user113600
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Evaluating $\int_m^\infty \frac {x^{v+r-1}}{(x^v-m^v+q)^{b+1}}dx$

Let $v,q,b ∈ \mathbb R^+ $ and $m∈ \mathbb R^+ ∪$ {0} , then how do we evaluate the definite integral $$\int_m^\infty \frac {x^{v+r-1}}{(x^v-m^v+q)^{b+1}}dx$$ , where $r ∈ \mathbb N $ ? At least a solution(exposition) for $r=1$…
Souvik Dey
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Question about a definite integral

We are doing definite integrals in university and I wanted to practice but this problem is giving me a hard time. The problem is to evaluate the following integral: $ \displaystyle \int_{0}^{2\pi} \frac{1}{5+4\cos(x)} dx$ For the antiderivative I…
PaulH
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Is there only numerical method to find this defenite integral or any other way?

Is there only numerical method to find this integral or any other way? Any references or links would be helpful. $$\int _{a} ^ {b} e ^ {-x^2/2} dx .$$
dexterdev
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Evaluation of $\displaystyle \lim_{u\rightarrow \infty}\frac{\int^{\pi u}_{1}\frac{\sin^2(5x)}{x}dx}{\ln(u^2+u^{-2})}$

Evaluation of $\displaystyle \lim_{u\rightarrow \infty}\frac{\int^{\pi u}_{1}\frac{\sin^2(5x)}{x}dx}{\ln(u^2+u^{-2})}$ What I Try: Using newton leibniz formula $\displaystyle \lim_{u\rightarrow \infty}\frac{\bigg(\int^{\pi…
jacky
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Tricky double integral

$$\int_{-1}^{1}\int_{-\sqrt{1-x^2}}^{\sqrt{1-x^2}}\sqrt{4-x^2-y^2} \, dy \, dx$$ Is there a way to calculate this definite integral by hand?
greg lee
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Evaluating $\arctan\left(\sin\frac12(\beta-x)\csc\frac12(\beta+x)\right)$ when $x=\pi$

I'm trying to evaluate the following: $$\frac{\sin x}{1-\cos\beta\cos x} - 2\cot\beta \arctan\left(\sin \left(\frac{\beta-x}{2}\right) \csc \left(\frac{\beta+x}{2}\right)\right)$$ This is the result of an integral and I need to evaluate at $x = \pi$…
rdemo
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Integrating $\int_0^\infty \frac{x\sin(tx)}{1+x^2} dx$ without contour integration?

How do I integrate this without contour integration? $$\int_0^\infty \frac{x\sin(tx)}{1+x^2} dx$$ I have tried everything, splitting the integral from $0$ to $1$ and $1$ to infinity and using the geometric series summation, rewriting $\sin(tx)$ as…
uggupuggu
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Clarification regarding $\int_0^{\frac{\pi}{2}} \tan^nx dx$

So, I was trying to calculate the integral $$\ I(n) = \int_0^{\frac{\pi}{2}} \tan^n x dx$$ I used the substitution $\sin^2x = t$. Rearranging, I got the expression: $$\ I(n) = \frac{1}{2} \cdot \int_0^1 t^{\frac{n-1}{2}} \cdot (1-t)^{\frac{-1-n}{2}}…
user1173615
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definite integral $\int_0^\pi\bigg(\frac{\sin(2x)\sin(3x)\sin(5x)\sin(30x)}{\sin x\sin(6x)\sin(10x)\sin(15x)}\bigg)^2dx$

Evaluate the definite integral $$\int_0^\pi\bigg(\frac{\sin(2x)\sin(3x)\sin(5x)\sin(30x)}{\sin x\sin(6x)\sin(10x)\sin(15x)}\bigg)^2dx$$ My Trial: Using double angle formula, $$\frac{\sin(2x)\sin(3x)\sin(5x)\sin(30x)}{\sin…
YANGyu
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Exact notation for an integral, evaluating from $u(a)$ to $u(b)$

I know how to do definite integrals and u-substitution. I am not clear on the notation for how to represent the lower and upper limits when switching variables in terms of $x$ to $u$ as shown in the examples below. I am looking for the exact…
William Garske
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Evaluate the integral ratio $\frac{\int_0^{\theta_0}\sin^2\theta\sin^2(u\sin\theta)d\theta}{\int_0^{\theta_0}\sin^2(u\sin\theta)d\theta}$

How to evaluate the ratio of integrals $$ I(u)=\frac{\int_0^{\theta_0}\sin^2\theta\sin^2(u\sin\theta)d\theta}{\int_0^{\theta_0}\sin^2(u\sin\theta)d\theta} $$ for the case when $\theta_0 \to 0$ and the case when $\theta_0 \to \frac{\pi}{2}$, where…
ecook
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