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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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$\int_{0}^{\infty}\frac{\ln x dx}{x^2+2x+2}$

$$\int_{0}^{\infty}\frac{\ln x .dx}{x^2+2x+2}$$ $$\int_{0}^{\infty}\frac{\ln x .dx}{x^2+2x+2}=\int_{0}^{\infty}\frac{\ln x .dx}{(x+1)^2+1}\\ =\ln x\int_{0}^{\infty}\frac{1}{(x+1)^2+1}-\int_{0}^{\infty}\frac{1}{x}\frac{1}{(x+1)^2+1}dx$$ and then lost…
mathspuzzle
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$\int\limits_{0}^{1}(\prod\limits_{r=1}^{n}(x+r))(\sum\limits_{k=1}^{n}\frac{1}{x+k})dx$

The value of $\int\limits_{0}^{1}(\prod\limits_{r=1}^{n}(x+r))(\sum\limits_{k=1}^{n}\frac{1}{x+k})dx$ is equal to $(A)n\hspace{1cm}(B)n!\hspace{1cm}(C)(n+1)!\hspace{1cm}(D)n.n!$ I…
diya
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Summation and integration

$$\sum_{r=1}^n\int_{-r(r!)}^{r(r!)}\frac{|\sin x|}{1+\pi^x}\,dx=a((n+b)!-c!),$$where $a,b,c\in \Bbb N$. Find the value of $a+b+c$. My attempt Let $$I=\sum_{r=1}^n\int_{-r(r!)}^{r(r!)}\frac{|\sin x|}{1+\pi^x}\,dx.\tag{1}$$ Apply…
Vinod Kumar Punia
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Evaluate $\int_{2}^{3}\frac{(x-2)^{100}}{(x-1)}dx$

Evaluate $$\int_{2}^{3}\frac{(x-2)^{100}}{(x-1)}dx$$ Is there an easy method to evaluate the integral. I wanted to try by using long division but I think that it will be very complicate. Any help ,thanks
E.H.E
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Why can't a value for this definite integral be found?

I was trying to find out if $\int _0^{\infty }\:\frac{1}{\left(1+x^3\right)^{\frac{1}{2}}}dx$ converges or diverges. I split it into a sum, that is $\int _0^{1 }\:\frac{1}{\left(1+x^3\right)^{\frac{1}{2}}}dx$ + $\int _1^{y…
2good4this
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Induction Problem using calculus.

Let $$ f_{o}(x)=\ln(x) \: \:\text{and} \: \:f_{n+1}(x) = \int_0^x\!f_n(t)\:\text{dt} $$ Prove by induction that $$ f_n(1) = - \frac{1}{n!} \sum_{i=1}^{n} \frac{1}{i} $$
JCMcRae
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Calculate $\int_{-\pi}^{\pi}\frac{1-\cos \frac{t}{4}}{5-2\cos t}dt$

Calculate the following integral: $$\int_{-\pi}^{\pi}\frac{1-\cos \frac{t}{4}}{5-2\cos t}dt$$ Any suggestions please? I do not really know where to start!
Mark
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Integral of the product of squared exponential and two erf functions

I'm trying to solve the following integral $$ \int_{-\infty}^{\infty} e^{-(\alpha t + \beta)^2}\operatorname{erf}(at + b)\operatorname{erf}(ct + d)\text{d}t $$ I've tried with differentiation under the integration sign and integration by parts with…
cdguarnizo
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Evaluate $\int_{-1}^{1}\frac{(x^{5}-x^{3}-x)(1+x^{2})^{\frac{1}{4}}\sin5x}{\sqrt{1-x^{2}}}dx$?

How to evaluate these integrals: $$\int_{-1}^{1}(x^{5}-x^{3}-x)(1+x^{2})^{\frac{1}{4}}\sin5xdx$$ $$\int_{-1}^{1}\frac{(x^{5}-x^{3}-x)(1+x^{2})^{\frac{1}{4}}\sin5x}{\sqrt{1-x^{2}}}dx$$ Can anybody help me ? Thank you!
Roin
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deriving general form of rational log integral

Is there a nice/clever method to derive a general closed form for: $$\displaystyle \int_0^1 \frac{\ln(1+x^a)}{1+x}dx, \;\ a>1\quad?$$ I thought maybe start with differentiating w.r.t. $a$. This gives $\displaystyle \int_0^1…
Cody
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A question regarding changing the $dx$ in an integral

How can I go from $\int_{-\infty}^{0} \dfrac{-kx}{(x^2+d^2)^{\frac{3}{2}}}dx$ to $\dfrac{-k}{2}\int_{-\infty}^{0} \dfrac{d(x^2+d^2)}{(x^2+d^2)^{3/2}}$ This is a step from my physics book and it doesn't explain it. Can you help?
user3601507
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Simplify $\int_{0}^{1}\ln(x-a)\ln x\,\mathrm{d}x$, for $a<0$

Let $a<0$. The following integral: $$\int_{0}^{1}\ln(x-a)\ln x\,\mathrm{d}x$$ can be computed to yield the…
a06e
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Calculate the integral without knowing the integrand

How can I calculate this integral? $$\int_0^{2014} \frac{f(x)}{f(x) + f(2014 - x)}dx$$
MathDav
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Definite integral: finding the function knowing the area from a to b?

Normally, we have a function $f(x)$ which we integrate in order to find the area under the curve from some a to some b.... but is it possible to solve an equation going in the other direction? Meaning that if I know the area under a given curve…
Dann
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Evaluating $\int_0^{\infty }{ \frac{(x^7)(1-x^{12})}{(1+x)^{28}}}dx$

Evaluate: $$\int_0^{\infty } {\frac{(x^7)(1-x^{12})}{(1+x)^{28}}}dx$$ The answer is zero, but I cannot seem to figure out the steps.
Ally
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