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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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Seeking $\int_{0}^{\infty}\left((1+x)^{-n}-\frac{\sin x}{x}\right)\frac{dx}{x}$

We are seeking to evaluate this integral $(1)$ $$\int_{0}^{\infty}\left((1+x)^{-n}-\frac{\sin x}{x}\right)\frac{\mathrm…
user569129
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Finding $ \int_{-\pi /2}^{\pi /2} \frac{\log (1 + b \sin x)}{\sin x}\,\mathrm dx$ given $|b|<1$

Find $$\int_{-\pi/2}^{\pi/2}\frac{\log(1+b\sin x)}{\sin x}\,\mathrm dx$$given that $|b|<1$. I split the integral into$$I=\int_0^{\pi/2}f(x)\,\mathrm dx+\int_{-\pi/2}^0f(x)\,\mathrm dx$$ For the second term made the substitution $x =-t$ and…
user619072
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How to derive $\int_0^1 \frac{\sin(\pi x)}{x^3-1}dx=\frac19\cosh\left(\frac{\sqrt{3}\pi}{2}\right)$?

I am interested in the integral$$\int_0^1 \frac{\sin(\pi x)}{x^3-1}dx=\frac19\cosh\left(\frac{\sqrt{3}\pi}{2}\right)$$ I thought about approaching this by expanding $\sin(\pi x)$ into its taylor series: $$I=\int_0^1 \frac{\sin(\pi…
aleden
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Find Value of $g'(0)$ if $g(x)$ is inverse of $f(x)$, where $f(x)=\int_{2}^{x}\frac{1}{\sqrt{(1+t^4)}}dt$.

Find Value of $g'(0)$ if $g(x)$ is inverse of $f(x)$ where $$f(x)=\int_{2}^{x}\frac{1}{\sqrt{(1+t^4)}}dt.$$ I had tried following things finding $f(x)$ by integration but failed is $g(x)$ is inverse of $f(x)$ then $$fog = x$$ So $$f'(g(x)) =…
Mohit
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Method to solve this integral with algebraic and trigonometric terms

I have the definite integral $\displaystyle\int_{0}^{\pi }\frac{x^2 \cos(x)}{(1+ \sin(x))^2}\,dx.$ Since there are both algebraic and trigonometric functions in the numerator, I don't know what substitution to make. Can someone tell the method of…
Abhijith S Raj
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a tough integral: $\int_{0}^{1}{x-3x^3+x^5\over 1+x^4+x^8}\ln(-\ln x)\mathrm dx={\pi \ln 2\over 6\sqrt{3}}$

How can we show that $(1)$ $$\int_{0}^{1}{x-3x^3+x^5\over 1+x^4+x^8}\ln(-\ln x)\mathrm dx={\pi \ln 2\over 6\sqrt{3}}?\tag1$$ Trying a substitution $$t=-\ln x, x=e^{-t}, \mathrm dx=-e^{-t}\mathrm dt$$ $$\int_{0}^{\infty}{e^{-2t}-3e^{-4t}+e^{-6t}\over…
user553457
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Calculating $\int^1_0\frac{x \log x}{1 + x^2} dx$

Trying to reinvent the spirit of Calculus, I was trying to Understand the Leibniz Rule of Differentiation under Integration sign. To check my understanding, I was considering this integral: $$ I(b) = \int^1_0 \frac{x \log(b + x)}{1 + x^2} dx…
sarker306
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Evaluating $\int_0^1 \sqrt{1-x^2}\cos(ax) dx$

I want to evaluate this integral $$I=\int_0^1 \sqrt{1-x^2}\cos(ax) dx \quad (a \in \mathbb{R})$$ but I cannot find a useful strategy. Could you please give me a hint?
Mai Hoàn Hảo
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$\displaystyle \int_0^{+\infty} \dfrac{\ln(x+1)}{x^2+1} \ \mathrm dx$

Find the value of: $$\int_0^{+\infty} \dfrac{\ln(x+1)}{x^2+1} \ \mathrm dx$$ I tried using substitution but it doesn't work. Is there any way to solve it ?
Youssef Khiari
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finding ratio of two definite integration

If $\displaystyle A = \int^{1}_{0}x^{\frac{7}{2}}(1-x)^{\frac{5}{2}}dx$ and $\displaystyle B = \int^{1}_{0}\frac{x^{\frac{3}{2}}(1-x)^{\frac{7}{2}}}{(x+3)^8}dx\;,$ then value of $AB^{-1} = $ Attempt: i have tried using gamma function $\displaystyle…
DXT
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Second Mean Value Theorem for Integrals, Variant

I want to prove the following theorem, which Wikipedia refers as 'Second Mean Value Theorem' Suppose that $g(x)$ is a non-negative monotonically decreasing function on the interval $[a, b]$, and its derivative is continuous. For $f(x)$ continuous…
zxcvber
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Worthwhile first-year-level definite-integrals-"proper"

If one mentions the topic of evaluating definite integrals without the fundamental theorem of calculus, I think of things like $$ \int_0^\infty \frac{\sin x} x \, dx \quad \text{ or } \quad \int_{-\infty}^\infty e^{-x^2}\,dx $$ i.e. "definite…
Michael Hardy
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Integration of $\int^\pi_0 \frac{dx}{1+\sin x}$

$$\int^\pi_0 \frac{dx}{1+\sin x}$$Multiplying numerator and denominator by $\sec x$ $$\int^\pi_0 \frac{\sec x \ dx}{\sec x+\tan x}$$Next, I multiplied numerator and denominator by $\sec x+\tan x$ which gives $$\int^\pi_0 \frac{(\sec^2x+\sec x\tan…
Dangnabbit
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Solving an integral

I am having trouble with this integral: $$\int_0^4\frac{x}{\sqrt{x^2+2x+25}}dx.$$ One obvious thing would be to complete the square: $x^2+2x+25=(x+1)^2+24$. But then, I don't know which substitution to use. Can anyone help? Thank you.
user30775
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$\int_{0}^{\infty}f\left(\frac{a}{x}+\frac{x}{a}\right)\frac{\ln x}{x}dx=\ln a\int_{0}^{\infty}f\left(\frac{a}{x}+\frac{x}{a}\right)\frac{dx}{x}$

Show that $\displaystyle \int_{0}^{\infty}f\left(\frac{a}{x}+\frac{x}{a}\right)\frac{\ln x}{x}dx=\ln a\int_{0}^{\infty}f\left(\frac{a}{x}+\frac{x}{a}\right)\frac{dx}{x}$ What substitution should i make for this.Both sides are looking alike,how to…
Brahmagupta
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