Mathematics Stack Exchange
Mathematics Stack Exchange

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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$\int_{0}^{\pi/2}\log(\sin^2\theta+k^2\cos^2\theta)d\theta=\pi\log\frac{1+k}{2},k\geq0$

Prove that $$\int_{0}^{\pi/2}\log(\sin^2\theta+k^2\cos^2\theta)d\theta=\pi\log\frac{1+k}{2},k\geq0$$ I tried but stuck in between. Let…
diya
  • 3,589
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Prove that $\int_{0}^{2\pi}\frac{x^2\sin x}{8+\sin^2x}=\frac{2\pi^2}{3}\ln\frac{1}{2}$

Prove that $$\int_{0}^{2\pi}\dfrac{x^2\sin x}{8+\sin^2x}=\frac{2\pi^2}{3}\ln\frac{1}{2}$$ My Attempt: $$\int_{0}^{2\pi}\dfrac{x^2\sin x}{8+\sin^2x}=\int_{0}^{2\pi}x^2\frac{\sin x}{8+\sin^2x}$$ I applied integration by parts,considering $x^2$ as…
user1442
  • 1,212
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$\int_{0}^{1}\frac{1-x}{1+x}.\frac{dx}{\sqrt{x+x^2+x^3}}$

$$\int_{0}^{1}\frac{1-x}{1+x}.\frac{dx}{\sqrt{x+x^2+x^3}}$$ My Attempt: $$\int_{0}^{1}\frac{1-x}{1+x}.\frac{dx}{\sqrt x\sqrt{1+x(1+x)}}$$ Replacing $x$ by $1-x$,we get $$\int_{0}^{1}\frac{x}{2-x}.\frac{dx}{\sqrt{1-x}\sqrt{1+(1-x)(2-x)}}$$ Then I got…
user1442
  • 1,212
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$\int\limits_{0}^{\pi/2}\frac{1+2\cos x}{(2+\cos x)^2}dx$

$\int\limits_{0}^{\pi/2}\frac{1+2\cos x}{(2+\cos x)^2}dx$ I tried to solve it. $\int\limits_{0}^{\pi/2}\frac{1+2\cos x}{(2+\cos x)^2}dx=\int\limits_{0}^{\pi/2}\frac{4+2\cos x}{(2+\cos x)^2}-\frac{3}{(2+\cos…
diya
  • 3,589
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$\int\limits_{-1/2}^{1/2}(\sin^{-1}(3x-4x^3)-\cos^{-1}(4x^3-3x))dx=$

$\int\limits_{-1/2}^{1/2}(\sin^{-1}(3x-4x^3)-\cos^{-1}(4x^3-3x))dx=$ $(A)0\hspace{1cm}(B)\frac{-\pi}{2}\hspace{1cm}(C)\frac{\pi}{2}\hspace{1cm}(D)\frac{7\pi}{2}$ I tried and got the answer but my answer is not matching the options given.Is my method…
diya
  • 3,589
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Will it be correct to intregrate when answer comes in via square roots since they can be positive or negative?

I tried to find the integral of the function $$\int_0^2x(x+2)^\frac{1}{2}dx$$ I substituted $$x+2=t^2$$ but while converting limits I found $t=2,-2$ as upper limit. When integrating with these different limits I get different answers while the…
Sikander
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What is wrong with this integration of $ \int_0^{2\pi}\sin x /(1 + A \sin x)$

A worked solution of the integral has been provided as an answer to a previous question. But I am still unclear why I get the wrong answer from the following method which uses a formula for the definite integral provided by Wolfram Alpha. Wolfram…
steveOw
  • 941
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Does this basic integral rule (property) hold true here?

Consider a function $f(x)$ which is undefined / indeterminate at $x=a$. Is it still true that, $$\int\limits_a^a f(a)\,\mathrm dx=0~?$$
asdsaf
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What's the area of one arch of a cycloid?

So, the cycloid is given with parametric equations: $$x=r(t-\sin{t})$$ $$y=r(1-\cos{t})$$ The teacher solved it like this: $$P=\int_a^by(x)dx$$ $x=x(t)$;…
A6SE
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Cyclic integration by parts trick

I have seen cyclic integration by parts trick used for trignometric integrals. I got the impression that it is necessary for the trick to work that derivative of the function is cyclic. However it is not so, because the trick works for polynomials…
Bezdomnyi
  • 139
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Hint for the integral by complex variable

Can anyone suggest a simple method to solve this integral by complex variables? $$\int_{0}^{2\pi }\frac{d\theta }{\omega -a \sin\theta }$$ where $|a|<|w|$. I am actually trying to find out the time period for non-uniform oscillator.
bpr3003
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Integral $\int \limits _0 ^\pi |\sin x + \cos x|\; dx$

$$\int \limits _0 ^\pi |\sin x + \cos x|\; dx$$ If I divide integral in two parts $\int\limits_0^{\frac{\pi}{2}}{(\sin x + \cos x)\,dx}$ and $\int\limits_{\frac{\pi}{2}}^\pi{(\sin x - \cos x)\,dx}$...I am getting $4$...Am I right?
user45799
  • 809
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Prove or disprove $\int_0^\infty {\frac{\sin ( 2 \omega x)}{{\sin x}}\frac{dx}{1 + x^2} = \frac\pi {e^2- 1}\frac{e^{2 \omega} - 1}{e^{2\omega - 1}}} $

I'd like to know how to prove or disprove that $$\int\limits_0^\infty {\frac{{\sin \left( {2 \omega x} \right)}}{{\sin x}}\frac{{dx}}{{1 + {x^2}}} = \frac{\pi }{{{e^2} - 1}}\frac{{{e^{2 \omega}} - 1}}{{{e^{2\omega - 1}}}}} $$ I always try to solve…
Pedro
  • 122,002
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Finding area between two curves, below and above x-axis

I'm trying to find the area between: $y = 2x^2 - 1$ and $y = x^2$ I have found that the intersection points are at $(-1,1)$ and $(1,1)$. But the part that confuses me is that $y = 2x^2 - 1$ goes below the x-axis and because of that I don't know how…
Stanko
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Complex definite Integration

Actually i don't know how to start with this problem. I have taken many substitutions but its becoming more and more complex. Please guide me to solve it. Thanks
Pratyush
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