Relating to integrations consisting of only(mainly) trigonometric functions and/or requiring substitutions by/of trigonometric functions.
Questions tagged [trigonometric-integrals]
1457 questions
2
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What's the integral of $\int \frac{1 - \cos kx }{1 - \cos x}dx$?
How do I evaluate $$\int \frac{1 - \cos kx }{1 - \cos x}dx$$?
I realise that $\cos kx$ can be expressed as a Chebyshev polynomial but because it's part of a fraction I can't see that this helps very much.
Ed Graham
- 123
1
vote
1 answer
Integral of rational trigonometric function
I would need some help trying to evaluate the following integral:
$$
I = \frac{1}{\pi}\int_0^{\pi} \frac{1-\cos \left(2n u\right)}{2 \cos \left(u\right)-x}\mathrm{d}u,
$$
where $|x|>2$ and $n=1,2,3,4,\dots$. However I have no idea how to proceed.
Matt
- 105
1
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1 answer
How to calculate this definite integral involving trigonometic intergration in a quater period
Calculate
$$\displaystyle\int\limits^{\cssId{upper-bound-mathjax}{\frac{{\pi}}{2}}}_{\cssId{lower-bound-mathjax}{0}} \dfrac{1}{1+\tan^n\left(x\right)}\,\cssId{int-var-mathjax}{\mathrm{d}x}$$
where $n$ stands for all natural numbers.
Bill
- 11
1
vote
4 answers
How do I solve a confusing trig substitution?
Hi I am trying to solve an integral problem that involves trig substitution. First I tried completing the square, which gave me $1/\sqrt{(x+3)^2+2^2}$. I know I am supposed to use $x = \arctan(\theta)$. Does that mean it should be:
$x + 3 =…
Niko H
- 53
1
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2 answers
why radians can be converted to reals in calculus?
Consider this integral:
$$ \int \sin^2x dx = \frac x2 - \frac {\sin2x}4 + C $$
Note the first term $\frac x2$ is a real as opposed to radian and can, in fact, be substituted with a real number when taking definite integral.
To make the statement…
Astrick Harren
- 163
1
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2 answers
Can a limit be changed twice when evaluating an integral?
I am asked to evaluated
$ \int^{\frac{3\sqrt3}{2}}_0 \frac{x^3}{(4x^2+9)^{\frac{3}{2}}} $
where $a>0$
the text book shows that during the u-substitution you should get
$\int^{3\sqrt3}_0 {\frac{{\frac{1}{2}}(u^3)}{\sqrt(u^2+9)^3}\frac{1}{2}}du…
C_bri
- 31
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2 answers
Definite Integral of sin(arccosh(x)+7)
How would I go about computing $\displaystyle\int_{10}^{16}\sin(\cosh^{-1}(x)+7)\mathrm dx$?
I haven't attempted anything yet, because I don't even know how to integrate the inverse hyperbolic cosine.
Saketh Malyala
- 13,637
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4 answers
Integrate $\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sec^2x}{\sqrt[3]{\tan\ x}}dx$
$$\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\frac{\sec^2x}{\sqrt[3]{\tan\ x}}dx$$
$$f(x) = (\tan \ x)^{\frac{2}{3}}, \ f'(x) = \frac{2}{3} \cdot (\tan \ x)^{-\frac{1}{3}} \cdot \sec^2x$$
$$\therefore…
StopReadingThisUsername
- 1,553
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vote
2 answers
Evaluate $\int_{-1/2}^{1/2}$ $1 \over \sqrt{1 - x^2}$ $dx$
I would like to evaluate
$\int_{-1/2}^{1/2}$ $1 \over \sqrt{1 - x^2}$ $dx$
Approach
We substitute $x = \sin u$ $\Rightarrow$ $dx \over du$ $=$ $\cos u \Rightarrow dx$ $=$ $\cos u$ $du.$
This leads to
$\int_{\sin(-1/2)}^{\sin(1/2)} 1$ $du$ =…
Julian
- 1,401
1
vote
1 answer
Evaluate $\int_{-1}^1T_n(x){1 \over \sqrt{1 - x^2}}\,dx$
Given
$T_n: [-1, 1] \rightarrow \Bbb R$ with $ n \in \Bbb N_0$,
$$ T_n(x) := \cos(n \arccos x)$$
I have to show that
$\displaystyle\int_{-1}^1 T_n(x){1 \over \sqrt{1 - x^2}}\,dx=\pi$ when $n = 0$,
and otherwise it's identical with $0$.
Approach
I…
Julian
- 1,401
1
vote
1 answer
Why is $\int \cos(n\arccos(\sin(x)) = \int \cos(nx)$ on $[-1, 1]$?
I would like to evaluate
$$\int \cos(n\arccos(\sin(x)) dx$$
with $x \in [-1, 1],$ and I used a calculator do to so. At the very beginning, the calculator simplifies
$$\int \cos(n\arccos(\sin(x)) dx = \int \cos(nx) dx.$$
This looks like…
Julian
- 1,401
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Integrating tan x without substitution.
A student asked how to integrate tan x and he was trying while using integration by parts.
I taught him the substitution method but he then asked why we can't use integration by parts.
My immediate thought was it is possible just much more…
David
- 1
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1 answer
How to easily visualise $ \int_{-\pi}^{\pi} \cos (nx)\cos(kx)dx=0$?
Is there an easy way to visualise that the following holds?
$$ \int_{-\pi}^{\pi} \cos (nx)\cos(kx)dx=0$$
I thought about simply switching the cosine to sine:
$$ \int_{-\pi}^{\pi} \cos (nx)\sin\left( kx+\frac{\pi}{2}\right)dx=0$$
Now its an even…
bananenheld
- 599
0
votes
0 answers
Is this a valid method in Trig Substitution to skip steps?
I've been practicing for a test tomorrow, and with the past few Trig Sub questions I have done involving $\sqrt{x^2 + a^2}$, $sec(u)$ always equals the square root in the expression.
Is this true for all values of $a$, or just $a = 1$? And if this…
Andog
- 1
- 1
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1 answer
Integral of $(sin(x))(sin(x))$
Math noob here, I work mainly on electronics and math is not my strongest suit.
I am trying to find out the type function of total energy transfered throughout a AC half wave. If any one of you is in electronics, it's for trying to linearize the…
sloth1089
- 11