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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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Prove: $\int_{0}^{x} f(u)(x-u) du = \int_{0}^{x} \left(\int_{0}^{u} f(t)dt\right)du$

Prove: $$\int_{0}^{x} f(u)(x-u) du = \int_{0}^{x} \left(\int_{0}^{u} f(t)dt\right)du$$ I tried to differentiate both sides but then I get this $$f(x)(x-x) = \int_{0}^{x} f(t)dt$$ Any help?
user349557
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Integration constant: What am I missing?

Deriving some physics formulas with my son, I managed to confuse myself. From: $$a_0 = \frac{dv}{dt} \implies a_0\, dt = dv \implies \int_{v_0}^{v} dv = \int_{t_0}^{t} a_0\, dt$$ we have: $$v=v_0 + a_0\Delta t \tag{1}$$ If $t_0 = 0$ we have: $$v=v_0…
blackened
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$\lim_{n\to\infty}\int_0^{\pi/2} \frac{\sin^n(x)}{1+x^2} \, dx$

$$\lim_{n\to\infty}\int_0^{\pi/2} \frac{\sin^n(x)}{1+x^2} \, dx$$ Is it right answer ? $$ \begin{cases} 0, & x \ne \pi/2\\[8pt] \dfrac{2\pi}{4 + \pi^2}, & x = \pi/2 \end{cases} $$
markovian
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finding definite integral involving inverse of cot function

finding $\displaystyle \int^{\pi}_{-\frac{\pi}{3}}\bigg[\cot^{-1}\bigg(\frac{1}{2\cos x-1}\bigg)+\cot^{-1}\bigg(\cos x - \frac{1}{2}\bigg)\bigg]dx$ Attempt: \begin{align} & \int^{\frac{\pi}{3}}_{-\frac{\pi}{3}}\bigg[\cot^{-1}\bigg(\frac{1}{2\cos…
DXT
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Is there any simple method to evaluate this integral $\int_{0}^{2m\pi} \frac{1}{\sin^4(x)+\cos^4(x)}d x?$

I want to evaluate the following definite interal \begin{gather*} \int_{0}^{2m\pi} \frac{1}{\sin^4(x)+\cos^4(x)}d x, \end{gather*} where $m$ is a given positive integer. I have calculated this integral. But my method is fairly complex. My method…
azc
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Find the following Definite Integral $\int_{-1}^{1}(\{x+1\}\{x^2+2\})+(\{x^2+3\}\{x^3+4\})dx$, where $\{.\}$ is the fractional part of $x$.

Integrate: $$\int_{-1}^{1}(\{x+1\}\{x^2+2\})+(\{x^2+3\}\{x^3+4\})dx$$ My attempt: We know that within the interval $[0,1)$, the following is true: $$\{x+1\}=x$$ $$\{x^2+2\}=x^2$$ $$\{x^2+3\}=x^2$$ $$\{x^3+4\}=x^3$$ Therefore…
Student
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Zero mean and unit variance for sum of two Gaussians

Given $f(x)$, a real function, expressed as the sum of two Gaussian functions: $$f(x)=\frac{\alpha}{\sqrt{2\pi\sigma_1^2}}\,\,e^{-\frac{(x-\mu_1)^2}{2\sigma_1^2}}+\frac{1-\alpha}{\sqrt{2\pi\sigma_2^2}}\,\,e^{-\frac{(x-\mu_2)^2}{2\sigma_2^2}}$$ where…
JFNJr
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An expression for the definite integral $I_n = \int_0^{\pi/4}{\tan^n{x}\,\mathrm{d}x}$

I have the following definite integral: $$I_n = \int_0^{\pi/4}{\tan^n{x}\,\mathrm{d}x}\quad ,\forall n \in \mathbb{N}$$ Calculate $I_0$ and $I_1$. Calculate $I_n + I_{n+2}$. Can we deduce $I_n$? Here is my solution: $$I_0 =…
user265759
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Change of limits in definite integration

The definite integral $\displaystyle\int_0^1\frac{\mathrm dx}{1+x^2}$ is evaluated as such: Letting $x=\tan\theta$, $\mathrm dx=\sec^2\theta\ \mathrm d\theta$, $\begin{cases}x=0\\\theta=0\end{cases}$,…
Kenny Lau
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Integrate by substitution method

Need help on solving integrals using substitution. As I have only solved ones with Newton-Leibniz, I don't know how to solve this types: $$\int_0^1 \frac{\sqrt{e^x}dx}{\sqrt{e^x+e^{-x}}}$$
nlimits
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Evaluation of these integrals

I need a hint to evaluate the integrals: $$ \int _{0}^{p}\tanh(\pi k)k\,dk\quad \text{and}\quad \int _{0}^{s-1/2}\tan(\pi k)k\,dk $$ Here $p$ and $s$ are real numbers. I know I can evaluate them by power series but how else can I evaluate these…
Jose Garcia
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Find $\int_{1}^{2}\frac{x-1}{x^2\sqrt{x^2+(x-1)^2}}$

Find $\int_{1}^{2}\frac{x-1}{x^2\sqrt{x^2+(x-1)^2}}$ I tried to solve it by using the property $\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx$ Let…
Vinod Kumar Punia
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Definite integral problem involving trigonometric functions

$$\int_0^{\frac{\pi}{4}}\frac{\sec x}{1+2\sin^2x}dx=?$$ Attempt: $$=\int_0^{\frac{\pi}{4}}\frac{\sec x}{1+(1-\cos 2x)}dx$$ $$=\int_0^{\frac{\pi}{4}}\frac{\sec x}{2-\cos 2x}dx$$ $$=\sqrt{2}\int_0^{\frac{\pi}{4}}\frac{1}{\sqrt{1+\cos 2x}(2-\cos…
Aditya Dev
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Definite integral on fractional part function .

Q) Let $ I_{1}=\displaystyle \int_{0}^{10^{4}}\frac{\{\sqrt{x}\}}{\sqrt{x}}dx$ and $I_{2}=\displaystyle \int_{0}^{10} x\{x^{2}\}$dx .Find $I_{1},I_{2}$ ? (here {.} denotes fractional part of $x$) My Attempt For Solving $I_{1}$ we i split…
Sujith Sizon
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Evaluating $\int_0^a \frac{x^{b}+x^{c}}{(1+x)^{b+c+2}}$

Let $0
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