# Tagged Questions

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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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### How to integrate this using u substitution? [closed]

How can I solve this using u substitution or trigonometry? $$\int e^{2x} \cos(e^{2x}) dx = \frac 1 2 \sin(e^{2x}) + \text{constant}$$
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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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### Trig Substitution Integration

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? $$3a^2\int_{0}^{2\pi} \sin^2(\theta)\cos^4(\theta) \, d\theta$$
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### Is there a way besides integration by parts to solve this integral?

$$\int_{0}^{2\pi} -10\cos^9(t)\sin^4(t)t^4\,dt$$ Maybe a formula for this form or something?
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### Trigonometric substitution for integral question.

I'm reviewing my quizzes to study for midterm tomorrow, and I came across a problem where I'm supposed to integrate: $$\int\frac{1}{x^2\sqrt{4-x^2}}dx$$ I used Mathematica to solve the problem and ...
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### Trigonometric integral with the function- sin

I need some help with this integral please: $\displaystyle\int x\sin\frac{1}{x}dx$
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### Integrating $\sqrt{1+\cos^2}$

As part of my Calculus II final we had a bonus question that had $$\int \sin(x)\sqrt{1+\cos^2(x)} \, dx \tag{*}$$ This set-up integral was not given and I know that (*) is easy to solve with ...
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### A tricky integral

I'm trying to find the exact value of $$\int_{\frac{1}{\sqrt{3}}}^{\sqrt{3}} \frac{\arctan{(x^2)} }{1+x^2} \, dx$$ Ostensibly, I'd want to use this: $$\frac{d}{dx}\arctan{(x)}=\frac{1}{1+x^2}$$ But ...
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### Why is the integral of $\cos(x) - \sin(x)$ from $0$ to $π/4$ equal to $\sqrt2 - 1$?

Why is $$\int_ {0}^{π/4} {\cos(x)} - {\sin(x)} \ \mathrm{d}x=\sqrt2 -1$$ This answer popped up on a problem I was doing and it piqued my interest. Can anyone help me out?
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### Odd $\sin/\cos$ integral

How to evaluate $$\int \frac{\sin^3 x}{\cos^5x}dx\ ?$$ I've tried various substitutions with $\sin x = u$ or $\cos x = u$, I've tried using Euler's formula which result in too heavy calculations and ...
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### Integral with goniometric functions $\int(1+\cos^2x-\sin^2x)dx$

I am solving this example: Transcription: \begin{align} &\int(1+\cos^2x-\sin^2x)dx=\int(1+1-\sin^2x-\sin^2x)dx=\int(2-2\sin^2x)dx=\\ ...
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### $\int\frac{1}{\sin(x-a)\sin(x-b)}\,dx$

I'm stuck in solving the integral of $\dfrac{1}{\sin(x-a)\sin(x-b)}$. I "developed" the sin at denominator and then I divided it by $\cos^2x$ obtaining ...
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### Homework help, Trigonometric intergral [closed]

Can someone please help me with this question. $$\int \ \frac{16}{1-\cos8x} \ \ dx \ \ .$$ Thank You
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### Reduction formula tricky problem (Further Maths: F3)

$\int \:e^{ax}\cos ^n\left(x\right)dx$ I just cannot get it to reduce, I keep ending up with too many species in the next integral to use parts again. I have important Further Pure F3 exam in a month ...
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### Why is $\displaystyle\int^{\infty}_{0}{(1-\cos x)\over{x^{2}}}dx = \frac\pi{2}$?

I have been having trouble understanding Fourier series and analysis in one of my classes. This is one problem from the text and we have to show that this is true. I have done other problems related ...