Questions tagged [indefinite-integrals]
Question about finding the primitives of a given function, whether or not elementary.
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What is the closed form of the indefinite integral of $(a - x)^n$ with respect to $x$?
A professor evaluated the following integral as follows
$$\int{(a-x)^3 dx} = -\frac{1}{4}(a-x)^4.$$ And I've seen this generalization online as well but when is it true? It doesn't hold for $n=1$ ...
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I need to convert $\frac{x-\arctan x}{x^2}$ into a power series, and then integrate it. The series should start at $n=1$. [closed]
I need to convert $\frac{x-\arctan x}{x^2}$ into a power series, and then integrate it. The series should start at $n=1$.
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Integrals of roots of rational functions
Is there a simple algorithm that allows one to compute the antiderivative of the n-root of an arbitrary rational function?
The solution can be expressed in terms of compositions, products, sums, of ...
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Find solution of the IVP $ y' = y+ \frac12 |\sin(y^2)|,\,\, x>0,\,\, y(0) = -1$
Consider the following initial value problem
$$ y' = y+ \frac12 |\sin(y^2)|,\,\,\,\,\,\, x>0,\,\,\, y(0) = -1$$
Which of the following statements are true?
1.) there exists an $\alpha \in (0,\infty)...
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Finding a general expression for the improper integral $\int_0^\infty K_1( ( k^2+\alpha^2)^{1/2}r)\sin(kz)\,\mathrm{d}k$
$\newcommand{\on}[1]{\operatorname{#1}}$
In solving a classical fluid mechanics problem involving flow in porous media, I encountered a delicate infinite integral ...
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Is there a nice closed form for the integral of $(\tan x)^{2n}$?
I just want to know if there is a nice closed form for this integral:
$$\int_{}^{}(\tan x)^{2n}dx$$
I know that a reduction formula exists and this is what I get by following it:
$$\begin{align*}
\int\...
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Integrals of the form $\int f\left(x,\frac{\sqrt{x+m}}{\sqrt{x+n}}\right)\,dx$
Let $b-a=m$ and $b+a=n$, where $a$ and $b$ are real numbers. Then the following identities hold:
$$\frac{1-e^{\pm\text{i}\alpha}}{1+e^{\pm\text{i}\alpha}}=\tan{\left(\frac12\sec^{-1}\left(\frac{x+b}{a}...
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Why is the 6 dropped from the final integral solution?
I was looking at this website to find an integral solution.
But as you can see in the picture the final solution lacks the $6$ that's present above.
Is this an error or the $6$ should really be ...
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Solving $\int \frac{7^{\frac 3{x^2}}-\sqrt[3]{x^{14}} -6}{2x^3} dx$ with Basic Integration Techniques [closed]
I'm trying to do this integral: $\int \frac{7^{\frac 3{x^2}}-\sqrt[3]{x^{14}} -6}{2x^3} dx$. But I get stuck here: $\int {7^{\frac 3{x^2}}\over{2x^3}} -{{\sqrt[3]{x^{14}}\over{2x^3}}} - {6\over{2x^3}} ...
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Why to use modulus in integration of $1/x$ [closed]
$$ \int \frac1x = \log_e |x|+C$$
Why is modulus sign needed. If this is because the domain of logarithmic function is $(0,\infty)$ Then why don't we mention the limitations of the domains of other ...
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$\int (\sin^6(x) + \cos^6(x))\,dx$ only using change of variables and basic trigonometric and integral properties
I'm stuck trying to solve this integral: $\int (\sin^6(x) + \cos^6(x))\,dx$. It was in an exam, I asked friends and even another teacher, but they didn't know how to solve it either, the catch is you ...
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Doubt on graph of indefinite integration starting from 0 instead of $-\infty$
The graph of integration of a function show us the area under the function.
If we specify the range of integration we plot the graph between the range.
So if we just use indefinite integration we ...
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Computing the indefinite integral $\int\frac{\sin^3(x)\cos^3(x)}{(1+\sin^2(x)\cos^2(x))^2}\text{d}x$
$$\int\frac{\sin^3(x)\cos^3(x)}{(1+\sin^2(x)\cos^2(x))^2}\text{d}x$$
My friend had challenged me to solve this question but I am unable to. As I have just began to study calculus, I am unable to ...
user1319345
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If a function is integrable and is of bounded variation, then must $\sum_{n\geq 1} f(n)$ converge?
Let $f:(0,\infty)\to [0,\infty)$ be a continuously differentiable function.
Assume that $\int_0^{\infty} f(x)dx<\infty$ and that $\int_0^{\infty}|f'(x)|dx<\infty$.
Claim: $\sum_{n=1}^{\infty}f(n)...
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I am facing problem in $\int (\sin(x)×\sin(2x)×\sin(3x)×...×\sin(nx))dx$ for $n\geq 4$ .
Q)I am facing problem in $$\int [\sin(x)×\sin(2x)×\sin(3x)×....×\sin(nx)]dx$$ for $n\geq 4$. I want a shortcut method to this Integral.
My Approach:
I can evaluate $\int [\sin(x)×\sin(2x)]dx$ and $\...
