# Questions tagged [trigonometric-integrals]

Relating to integrations consisting of only(mainly) trigonometric functions and/or requiring substitutions by/of trigonometric functions.

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### does different domain for inverse trigonometric function give different definite integral?

$\int_{-1}^1 {\frac{\tan^{-1}x}{1+x^2}} \; dx$ becomes $\int_{\tan^{-1}-1}^{\tan^{-1}1} {\theta} \; d\theta$ by substituting $x = \tan(\theta)$ different domain $\left(-\frac\pi2, \frac\pi2\right)$ or ...
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### Generalization of the result of $\int_0^{\infty} \frac{e^{-x^2} \sin \left(x^2\right)}{x^2} d x$.

When I came across the integral $$\int_0^{\infty} \frac{e^{-x^2} \sin \left(x^2\right)}{x^2} d x,$$ I didn’t know how to deal with it. After struggling, I thought of Feynman’s trick and Euler formula ...
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### Evaluate the following trigonometric integral with exponential function

Find the value of $$\int_0^\pi\mathrm{e}^{\mathrm{e}^{\cos\left(x\right)}}\cos\left(\sin\left(x\right)\right)\cos\left(\mathrm{e}^x\sin\left(\sin\left(x\right)\right)\right)dx$$ How to solve this ...
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### discrete least squares error with trigonometric polynomial

Hi I am studying about Trigonometric polynomial that minimize Discrete Least Squares Approximation. I found a related lecture note online but I got lost in one of the steps on page 27. I still not ...
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### prove orthonormal with respect to discrete inner product

I'm currently reviewing my textbook and I'm having trouble understanding a part here. While it seems intuitively true, I find it hard to prove them in the first place. I've tried working through the ...
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### After trigonometric substitution, writing the antiderivative in terms of $x.$

The following integral suggests trigonometric substitution $x=4 \sin (\theta)$ : $$\int \frac{x^2}{\left(16-x^2\right)^{3 / 2}} d x \text {. }$$ After making this substitution and integrating, we ...
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### Make an appropriate trigonometric substitution to rewrite the given integrand as an integrand in the angle $\theta$ containing no square roots. [duplicate]

Assume all trigonometric functions are positive. Remove all square roots. Do NOT Evaluate the integral completely. My integral is: $$\int \frac{\sqrt{49+x^2}}{x^2}\,dx$$ The only substitution I'm ...
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### Unable to crack $\int_0^{\frac1{\sqrt3}} \frac{\cot^{-1}\sqrt{2-x^2}}{1+x^2}dx=\frac{\pi^2}{30}$

I am unable to solve the integral $$\int_0^{\frac1{\sqrt3}} \frac{\cot^{-1}\sqrt{2-x^2}}{1+x^2}dx=\frac{\pi^2}{30}$$ after a number of attempts, except with a few observations below. 1). Despite the ...
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### Bessel function of the first kind of order zero in integral representation.

I'm studying alternative methods for elliptic boundary conditions. I picked the formula of Bessel function from this site https://dlmf.nist.gov/10.9 I'm looking for any available approach to solve ...
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### Half angle sine substitution?

After learning the Weierstrass substitution, I wondered what would happen if I tried doing a half angle sine substitution. Using the formula $$\sin\frac{x}{2}=\pm\sqrt{\frac{1-\cos x}{2}}$$I derived ...
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### Integral of $\arccos$ function combined with exponontial function

Is it possible to calculate this integral? $$\int_0^1 x^{j}\ e^{ax^2}\arccos(x)\ dx,\qquad a\in\mathbb{R}_+,\ j\in\mathbb{N}.$$ Although the integral looks neat and fairly simple in form. I tried ...
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### Integral of $\,\tan^{2n}(x)\,\mathrm dx$

I want to evaluate $\,\displaystyle I_{n}=\int_{0}^{\frac{\pi}{4}} \tan^{2n}(x)\,\mathrm dx$. I proved that $\,I_{n}+I_{n-1}=\dfrac{1}{2n-1}\,,\,$ where $I_{0}=\dfrac{\pi}{4}$. From that I found that (...
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