Tagged Questions

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Evaluate$\int_0^{\frac{\pi}{2}} \ln(1+\cos x) dx$

Find the value of the integral $\int_0^{\frac{\pi}{2}} \ln(1+\cos x)$ I tried putting $1+ \cos x = 2 \cos^2 \frac{x}{2}$, but am unable to proceed further. I think the following integral can be ...
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Integral of Sinc Function Squared Over The Real Line

I am trying to evaluate $$\int_{-\infty}^{\infty} \frac{\sin(x)^2}{x^2} dx$$ Would a contour work? I have tried using a contour but had no success. Thanks. Edit: About 5 minutes after posting this ...
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Evaluate $\int{\sin^3(x)\cos^2(x)}dx$

I'm trying to solve $\int{\sin^3(x)\cos^2(x)}dx$. I got $-\frac{1}{2}\cos(x)+C$, but the memo says $\frac{1}{5}\cos^5(x)-\frac{1}{3}\cos^3(x)+C$ This is my working: Your help is appreciated!
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Integration of $1/\sin^3 x$

I need a explanation of this problem: $$\int \frac{1}{\sin^3 x}\,dx$$ Change the variable $$t = \tan (x/2)$$ With use of $\tan$, $\cos$, $\sin$ and $\cot$, only. So how do I ...
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Dealing with absolute values after trigonometric substitution in $\int \frac{\sqrt{1+x^2}}{x} \text{ d}x$.

I was doing this integral and wondered if the signum function would be a viable method for approaching such an integral. I can't seem to find any other way to help integrate the $|\sec \theta|$ term ...
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The limit of Riemann sums $\sum_{k=1}^{n}\cos(\frac{k\pi}{2n})\frac{ \pi}{2n}$

Find the limit of Riemann sums $$\lim_{n\rightarrow \infty} \sum_{k=1}^{n}\cos(\frac{k\pi}{2n})\frac{ \pi}{2n}$$ on the interval $$[0,\frac{\pi}{2}]$$ Progress All I have managed to do is ...
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Indefinite integral of trignometric function

What is the trick to integrate the following $$\int \frac{1-\cos x}{(1+\cos x)\cos x}\ dx$$
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A closed form for $\int_{0}^{\pi/2}\frac{\ln\cos x}{x}\mathrm{d}x$?

The following integrals are classic, initiated by L. Euler. \begin{align} \displaystyle \int_{0}^{\pi/2} x^3 \ln\cos x\:\mathrm{d}x & = -\frac{\pi^4}{64} \ln 2-\frac{3\pi^2}{16} ...
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Evaluate $\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx$ [duplicate]

As the title shown, how to evaluate the indefinite integral $$\int\left({\frac{\arctan x}{\arctan x-x}}\right)^2 \,dx\ ?$$ Thanks.
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Duo Fresnel-like integrals $(??)$

I really wonder how I can prove the following integrals. $$\int_0^\infty \sin ax^2\cos 2bx\, dx=\frac{1}{2}\sqrt{\frac{\pi}{2a}}\left(\cos \frac{b^2}{a}-\sin\frac{b^2}{a}\right)$$ and ...
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Finite integral with goniometric functions, $\int_0^{\infty} \frac{8\sin^4(\pi f t)\tan^2(\pi f/2)}{(\pi^4 \tau^2 f^3) }df$

I have difficulties trying to find an algebraic solutions of the following integral: The $\tau$ in this formula is an integer, which is a very important fact because only then this integral is ...
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Antiderivative of $\frac{1}{1+\sin {x} +\cos {x}}$

How do we arrive at the following integral $$\displaystyle\int\dfrac{dx}{1+\sin {x}+\cos {x}}=\log {\left(\sin {\frac{x}{2}}+\cos {\frac{x}{2}}\right)}-\log {\left(\cos {\frac{x}{2}}\right)}+C\ ?$$
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A closed form for $\int_{0}^{\pi/2} x^3 \ln^3(2 \cos x)\:\mathrm{d}x$

We already know that \begin{align} \displaystyle & \int_{0}^{\pi/2} x \ln(2 \cos x)\:\mathrm{d}x = -\frac{7}{16} \zeta(3), \\\\ & \int_{0}^{\pi/2} x^2 \ln^2(2 \cos x)\:\mathrm{d}x = ...
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Evaluate: $I = \int^{\pi/2}_0 (\sqrt{\sin x}+\sqrt{\cos x})^{-4}dx$

Evaluate : $$I = \int_{0}^{\Large\frac\pi2} (\sqrt{\sin x}+\sqrt{\cos x})^{-4}\ dx$$ Attempt : \begin{align} I&=\int_{0}^{\Large\frac\pi2} (\sqrt{\sin x}+\sqrt{\cos x})^{-4}\ dx\\ ...
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Integration practice of $\int \frac{\sqrt{25-y^2}}{y}dy$

I need to solve $\int \frac{\sqrt{25-y^2}}{y}dy$. I originally thought IBP, but that led to a very large and confusing algebra problem. Then I started to look at the $\sqrt{25-y^2}$ and started to ...
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if $f(x)=x+\cos x$ then find $\int_0^\pi (f^{-1}(x))\text{dx}$?

I would be interest to show : if $f(x)=x+\cos x$ then find $\int_0^\pi (f^{-1}(x))\text{dx}$ ? my second question that's make me a problem is that : what is :$f^{-1}(\pi)$ ? I would be ...
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integration by parts of trig functions

Can anyone help me with this integral? $\int{x^3 \sin(x^4) dx}$ I set $u=x^3$, and I let $v=-\cos(x^4)$, so that $\frac{dv}{dx}=\sin(x^4)$ I tried using integration by parts, but, whenever I come ...
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Line integral of a vector function involving sine and cosine

I have line integral of a vector function: $\vec{F}=-e^{-x}\sin y\,\,\vec{i}+e^{-x}\cos y\,\,\vec{j}$ The path is a square on the $xy$ plane with vertices at $(0,0),(1,0),(1,1),(0,1)$ Of course it is ...
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Evaluate $\int_{0}^{\pi}\sin^5{\theta}\cos^2{\theta}\ d\theta$ [duplicate]

I'm trying to find the mass of a spherical object with a given density function, and to do so I must solve this integral $$\int_{0}^{\pi}\sin^5{\theta}\cos^2{\theta}\ d\theta,$$ but no matter which ...
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Double integral compute

I'm struggling with this one for a week: There is a range $R$ that it's points $(x,y)$ are defined as: For each $0 \le x \le 32$, all the values of $y$ are $\sqrt[5]{x} \le y \le 2$. We need to ...
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Closed form of a trigonometric integral sought

I am trying to evaluate the definite integral $I(a,b)$, with $a,b\in\mathbb{R}$, defined by $$I(a,b):=\int_{0}^{2\pi}\sqrt{1-(a+b\cos{\theta})^2}\mathrm{d}\theta.$$ Assume $a,b$ are suitably ...
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Evaluate the integrals $\int \sin{x} \cot^2{x} \,dx$ and $\int \cos{x} \cot^2{x} \,dx$.

Can you please show how to evaluate the integrals $$\int \sin{x} \cot^2{x} \,dx$$ and $$\int \cos{x} \cot^2{x} \,dx.$$
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Is this integral with sine and cosine such a challenge?

...or maybe I just don't know some specific trick with trigonometric functions? Well, anyway, here it is: $$\int{\sin^6{x}\cos^4{x}\, dx}$$ I'm bored with it, because I get 9 integrals out of 1 ...
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Compute $\int_0^1 \frac{\arcsin(x)}{x}dx$

$$\int_0^1 \frac{\arcsin(x)}{x}dx$$ This is a proposed for a Calculus II exam, and I have absolutely no idea how to solve it. Tried using Frullani or Lobachevsky integrals, or beta and gamma ...
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Logarithm and “basic” functions.

To express the antiderivatives of $\frac{1}{x}$, we cannot apply the formula $\int x^n dx=\frac{x^{n+1}}{n+1}+C$ and we need to introduce a new function, the logarithm. But how can we prove that ...
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How to integrate this formula with secant, exponential, and tangent?

How to integrate this? $$\int \sec^2(3x)\ e^{\large\tan (3x)}\ dx$$
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Convolution integral $\int_0^t \cos(t-s)\sin(s)\ ds$

How can I calculate the following integral? $$\int_0^t \cos(t-s)\sin(s)\ ds$$ I can't get the integral by any substitutions, maybe it is easy but I can't get it.
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A Hard integral in 2-D.

I'm having a trouble integrating (in $\mathbb{R}^2$) the following formula: $$\frac{t}{|B(x,t)|}\int_{B(x,t)} \frac{||y||}{(t-||x-y||^2)^{\frac{1}{2}}} dy$$ where $B(x,t)$ is the ball with center ...
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Integration of $x/\sqrt{x^2-7}$ using trigonometric substitution

Didn't find this one here so I'm asking away: I tried integrating by substitution by ended up with just $x + C$ which is clearly wrong. My work, as requested:
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Integration of $\int_{0}^{\frac{1}{2}}\frac{\sin^{-1}(x)}{\sqrt{1-x^2}} dx$ ??

I was solving the integration of inverse trigonometric function and faced a question which i find it hard to understand. I need to find the definite integration of this function. ...
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Definite Trig Integrals: Changing Limits of Integration

$$\int_0^{\pi/4} \sec^4 \theta \tan^4 \theta\; d\theta$$ I used the substitution: let $u = \tan \theta$ ... then $du = \sec^2 \theta \; d\theta$. I know that now I have to change the limits of ...
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Integrating quotients with polynomials in numerator and denominator that are raised to fractional powers

I'm working through a paper on momentum in electrodynamics that requires the integration below and would greatly appreciate any help. I'm pretty sure it evaluates to $2/d$ but I can't quite figure ...
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Trig integral with sine and cosine

What sort of formulas can I use to reduce this into something I can work with? $$3a^2\int_{0}^{2\pi} \sin^2(\theta)\cos^4(\theta) \, d\theta$$