Recent questions in Applications of integrals

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Integral CalculusAnswered question

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Integral CalculusAnswered question

Salvador Bush 2022-07-03

integral substitution not by u-sub

$\int \frac{1}{({x}^{2}+4x+9)}\phantom{\rule{thinmathspace}{0ex}}dx$

$\int \frac{1}{({x}^{2}+4x+9)}\phantom{\rule{thinmathspace}{0ex}}dx$

Integral CalculusAnswered question

Augustus Acevedo 2022-07-03

I have to integrate this : $\int \frac{{\mathrm{cos}}^{2}(x)}{1+\mathrm{tan}(x)}dx$

Integral CalculusAnswered question

hawatajwizp 2022-06-26

How to calculate the integral ${\int}_{0.5}^{1}\frac{1}{\sqrt{2x-{x}^{2}}}$?

Integral CalculusAnswered question

Ezekiel Yoder 2022-06-24

Primitive of a function with $\mathrm{sin}\frac{1}{x}$

I have the next integral:

$\int {\textstyle (}\frac{\mathrm{sin}\frac{1}{x}}{{x}^{2}\sqrt{(4+3\mathrm{sin}\frac{2}{x})}}{\textstyle )}\phantom{\rule{thinmathspace}{0ex}}dx,\phantom{\rule{thickmathspace}{0ex}}x\in {\textstyle (}0,\mathrm{\infty}{\textstyle )}$

I used the substitution $u=\frac{1}{x}$ and I got

$-\int {\textstyle (}\frac{\mathrm{sin}u}{\sqrt{(4+3\mathrm{sin}2u)}}{\textstyle )}\phantom{\rule{thinmathspace}{0ex}}du$

Can somebody give me some tips about what should I do next, please?

I have the next integral:

$\int {\textstyle (}\frac{\mathrm{sin}\frac{1}{x}}{{x}^{2}\sqrt{(4+3\mathrm{sin}\frac{2}{x})}}{\textstyle )}\phantom{\rule{thinmathspace}{0ex}}dx,\phantom{\rule{thickmathspace}{0ex}}x\in {\textstyle (}0,\mathrm{\infty}{\textstyle )}$

I used the substitution $u=\frac{1}{x}$ and I got

$-\int {\textstyle (}\frac{\mathrm{sin}u}{\sqrt{(4+3\mathrm{sin}2u)}}{\textstyle )}\phantom{\rule{thinmathspace}{0ex}}du$

Can somebody give me some tips about what should I do next, please?

Integral CalculusAnswered question

Davon Irwin 2022-06-21

Is there any short method to solve the integral

$I=\int \frac{\mathrm{sin}3x\cdot \mathrm{sin}\frac{5x}{2}}{\mathrm{sin}\frac{x}{2}}dx$

$I=\int \frac{\mathrm{sin}3x\cdot \mathrm{sin}\frac{5x}{2}}{\mathrm{sin}\frac{x}{2}}dx$

Integral CalculusAnswered question

Brajesh Kumar2022-06-16

Evaluate double integral of ${\int}_{0}^{1}{\int}_{0}^{x}{e}^{\frac{x}{y}}dxdy$

Integral CalculusAnswered question

juanberrio8a 2022-06-16

Heaviside and trig function integral $\int \mathrm{sin}(3t)\theta (t)dt$

Integral CalculusAnswered question

gnatopoditw 2022-06-14

How do I integrate $\frac{{x}^{2}}{\sqrt{9{x}^{2}+15}}$?

Integral CalculusAnswered question

hanglutuupx6 2022-06-02

Added angle formula to solve this indefinite integral $\int \frac{2\mathrm{cos}x-\mathrm{sin}x}{3\mathrm{sin}x+5\mathrm{cos}x}\phantom{\rule{thinmathspace}{0ex}}dx$

Integral CalculusAnswered question

Rachel Villa 2022-05-29

Evaluate the definite integral using trig. sub $\int \frac{{x}^{3}}{\sqrt{9-{x}^{2}}}dx$

Integral CalculusAnswered question

Davin Fields 2022-05-26

I do not understand why for the two integrals below, the result is always 0 unless $|k|=n$ , in which case the result is $\pi /2$

${\int}_{0}^{\pi}d\theta \mathrm{cos}k\theta \mathrm{cos}n\theta \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{\int}_{0}^{\pi}d\theta \mathrm{sin}k\theta \mathrm{sin}n\theta $

I have tried using trigonometric identities to get a general solution but I had no luck understanding the nature of the integrals. If anyone can point out any hints or patterns, it would be much appreciated.

${\int}_{0}^{\pi}d\theta \mathrm{cos}k\theta \mathrm{cos}n\theta \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{\int}_{0}^{\pi}d\theta \mathrm{sin}k\theta \mathrm{sin}n\theta $

I have tried using trigonometric identities to get a general solution but I had no luck understanding the nature of the integrals. If anyone can point out any hints or patterns, it would be much appreciated.

Integral CalculusAnswered question

Alisa Durham 2022-05-15

Antiderivative of a trigonometric integral

$\int \frac{{\mathrm{sin}}^{3}(x)}{({\mathrm{cos}}^{4}(x)+3{\mathrm{cos}}^{2}(x)+1)\cdot \mathrm{arctan}(\mathrm{sec}(x)+\mathrm{cos}(x))}$

I am unable to manipulate this integral. The actual integral in question is definite with limits from 0 to $\frac{\pi}{2}$, but I feel it can't be calculated without knowing the antiderivative.

My initial try was to take $\mathrm{sec}(x)+\mathrm{cos}(x)$ as t, but It didn't solve.

$\int \frac{{\mathrm{sin}}^{3}(x)}{({\mathrm{cos}}^{4}(x)+3{\mathrm{cos}}^{2}(x)+1)\cdot \mathrm{arctan}(\mathrm{sec}(x)+\mathrm{cos}(x))}$

I am unable to manipulate this integral. The actual integral in question is definite with limits from 0 to $\frac{\pi}{2}$, but I feel it can't be calculated without knowing the antiderivative.

My initial try was to take $\mathrm{sec}(x)+\mathrm{cos}(x)$ as t, but It didn't solve.

Applications of integrals are one of the popular but still complex fields that many students encounter not only in Math and Physics but in subjects like Programming and Engineering. For example, start with the applications of double integrals and you will see how it is useful in the construction of some building where the shape must be found or the gravity matters should be presented as the model. At the same time, if your questions are based on equations, consider geometric applications of definite integrals help that is also present below as you take a look through the relevant answers.