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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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$I=\int_{-1}^{2}\frac{xf(x^2)}{2+f(x+1)}dx$

Let $f:R\to R$ be a function defined by $f(x)=\begin{cases} [x] & x\leq 2 \\ 0 & x>2 \\ \end{cases}$ where $[x]$ is the greatest integer less than or equal to $x$.Then find $I=\int_{-1}^{2}\frac{xf(x^2)}{2+f(x+1)}dx$ This…
user1442
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how to evaluate $\int_{0}^{+\infty} e^{-at}(sin(t))^{n} dt$

How to evaluate the following definite integral : $$\int_{0}^{+\infty} e^{-at}\left(\sin(t)\right)^{n}\,dt$$
abd elkarim
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Definite integral of $\frac{1-x}{x}$ vs $\sqrt{\frac{1-x}{x}}$

I found myself intrigued by the result of the definite integrals of $\frac{1-x}{x}$ and $\sqrt{\frac{1-x}{x}}$ respectively. Both functions are seemingly similar, with vertical asymptotes at $x=0$. However the area bounded between the two and the…
Milo Wielondek
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Find the area bounded by the curves $y=-\sqrt{-x}$ and $x=-\sqrt{-y}$,where $x,y\leq 0$

Find the area bounded by the curves $y=-\sqrt{-x}$ and $x=-\sqrt{-y}$,where $x,y\leq 0$. I found the area $A=\int_{-1}^{0}-x^2+\sqrt{-x} dx=-1$ when we take its absolute value,it becomes $+1$ but the answer given in my book is $\frac{1}{3}$.Have i…
diya
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Method for solving $\int_0^1 \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^\alpha (1-x)^\beta dx$

I really don't know where to start in integrating this $\int_0^1 \frac{\Gamma(\alpha+\beta)}{\Gamma(\alpha)\Gamma(\beta)} x^\alpha (1-x)^\beta dx$
Ryan
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Prove that $\int_{0}^{1}\frac{\log(1+x)}{1+x^2}\, dx=\frac{\pi}{8}\log 2$

Prove that $\int_{0}^{1}\frac{\log(1+x)}{1+x^2}\, dx=\frac{\pi}{8}\log 2$ I have tried to sole the problem, but failed. Any one can help me to sole the problems. Here, I have tried to solve this using integration by parts : Let…
MKSar
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Check if an integral is convergent and calculate its value

I have improper integral $$\int_0^\pi \frac{\cos(x)}{(1-2\sin(x))^{1/3}} dx$$ I have to check if it is convergent. If yes, then i have to evaluate it. I think it is convergent. I can make substitution $t=\sin(x)$. But what next?
adm34
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Find the area bounded by two functions $f(x)=(\cos^{-1}|\cos x|)^2$ and $\cos^{-1}|\cos x|$ for the ordinates $|x|=2\pi$

Find the area bounded by two functions $f(x)=(\cos^{-1}|\cos x|)^2$ and $\cos^{-1}|\cos x|$ for the ordinates $|x|=2\pi$ I tried to solve this problem but could not get correct answer. I drew their…
diya
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Define g(x) as a function of $x$

Let $f(x)= \begin{cases} -1 & ,-2\leq x\leq 0 \\ \\ |x-1| & ,0
Brahmagupta
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Prove that $\int_{0}^{\infty}\frac{dx}{1+x^n}=\int_{0}^{1}\frac{dx}{(1-x^n)^{1/n}}$

Prove that $\int_{0}^{\infty}\frac{dx}{1+x^n}=\int_{0}^{1}\frac{dx}{(1-x^n)^{1/n}}$,where $n>1$ My Attempt: $\int_{0}^{1}\frac{dx}{(1-x^n)^{1/n}}=\int_{0}^{1}\frac{dx}{x(x^{-n}-1)^{1/n}}=\int_{0}^{1}\frac{x^{-n-1}dx}{x^{-n}(x^{-n}-1)^{1/n}}$ Now put…
user1442
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$\int_{0}^{\pi}|\sqrt2\sin x+2\cos x|dx$

$\int_{0}^{\pi}|\sqrt2\sin x+2\cos x|dx$ MyAttempt $\int_{0}^{\pi}|\sqrt2\sin x+2\cos x|=\int_{0}^{\pi/2}\sqrt2\sin x+2\cos x dx+\int_{\pi/2}^{\pi}|\sqrt2\sin x+2\cos x| dx$ I could solve first integral but in second one,i could not judge the mod…
user1442
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The integral of exponential function and modified Bessel function

everyone, I have an integral of exponential function and modified Bessel function to calculate, as follows $Q=\int\limits_0^\infty {z\exp \left( { - a{z^2}} \right){I_0}\left( {bz} \right)} dz$, where ${I_0}\left( \right)$ is the 0th order…
junxv penn
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$\int\limits_{1/2}^{2}\frac{1}{x}\sin (x-\frac{1}{x})dx$

$\int\limits_{1/2}^{2}\frac{1}{x}\sin (x-\frac{1}{x})dx$ has value equal to $(A)0\hspace{1cm}(B)\frac{3}{4}\hspace{1cm}(C)\frac{3}{4}\hspace{1cm}(D)2 $ I tried to solve this question by putting $x-\frac{1}{x}=t$ and limits have changed to…
diya
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Distance covered by integrating the Velocity of a Body

For a National Board Exam Review The velocity of a body is given by v(t) = sin(pi*t) where the velocity is given in meters per second and t is given in seconds. The distance covered in meters between t = 0.25 and t = 0.5. Answer is…
james
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Write integral of sin as a multiple

Our professor gave us this question. Write $\int^{2\pi}_{0}sin^{100}(x)dx$ as a multiple of $\int^{2\pi}_{0}sin^{98}(x)dx$ using simple techniques (like substitution or integration by parts). Can somebody give me a start on how to go about it?
rahulgarg12342
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