# Tagged Questions

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### Solving using integrating factor [on hold]

Q) Solve $y' = 2x + y$ using the integrating factor. Can anyone guide me with steps here? Help appreciated. Thanks.
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### Evaluating sums and integrals using Taylor's Theorem

Taylor's theorem states that $$f(x)-\sum_{k=0}^n\frac{f^{(k)}(a)}{k!}x^k = \int_a^x \frac{f^{(n+1)} (t)}{n!} (x - t)^n \, dt$$ This could be used to evaluate partial sums using knowledge of the ...
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### Calc 2: Integration by Parts w/ trig identities

$$\int e^{3\theta}\sec^4(e^{3\theta})\tan^{11}(e^{3\theta})d\theta$$ I just want to make sure that I'm doing this correctly so that I can understand the material. I would also appreciate any tips or ...
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### Is there a formal proof of this basic integral property?

This has really been bothering me because everywhere I have looked the answer has been "A proof has been omitted because the theorem is very intuitive" or "Proofs are very complicated and not worth ...
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### Looking for advice with the following integral

I have the following integral to evaluate: $$\frac{1}{f(t)}\int_0^t t^m (t + n)^o \sin(pt) \mathrm{d}t \quad m,n,o,p \in \mathbb{R}$$ I'm unable to proceed with this integral as it is non-trivial. ...
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### Numerical value of $\int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy$

Could somebody give me a numerical value for this integral? $$I = \int_0^1 \int_0^1 \frac{\arcsin\left(\sqrt{1-s}\sqrt{y}\right)}{\sqrt{1-y} \cdot (sy-y+1)}\,ds\,dy$$
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### Computing in closed form $\sum_{n=1}^{\infty}\frac{\operatorname{Ci}\left(\frac{3}{4}\zeta(2) \space n\right)}{n^2}$

What tools would you recommend me for computing the series below? $$\sum_{n=1}^{\infty}\frac{\operatorname{\displaystyle Ci\left(\frac{3}{4}\zeta(2) \space n\right)}}{n^2}$$ I lack the starting ...
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### Two methods to integrate?

Are both methods to solve this equation correct? $$\int \frac{x}{\sqrt{1 + 2x^2}} dx$$ Method One: $$u=2x^2$$ $$\frac{1}{4}\int \frac{1}{\sqrt{1^2 + \sqrt{u^2}}} du$$ ...
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### Problem with simplifying before integration

Can someone explain to me how did the du = 6y^(-1/3)dy went into the last equation?
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### Closed-form of $\displaystyle\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)$

Does the following series have a closed-form $$\sum_{n=1}^\infty\frac{(-1)^{n+1}}{n}\Psi_3(n+1)$$ where $\Psi_3(x)$ is the polygamma function of order 3. Here is ...
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### integration involving imaginary terms

How do we integrate forms of following type with imaginary terms involved? Can we get a closed form of it as result? ...
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### Integration using exponent

What could be the techniques we need to use to solve this integration $\displaystyle \int\tan^2\theta\frac{\sin^2(\sec\theta\tan\theta)}{\sec^2\theta}d\theta \tag1$? How do I convert this in to a ...
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### Integration with quadratic square root

What could be the techniques we need to use to solve this integration $\int\dfrac{s^2\sin^2\left(s\sqrt{ as^2+bs+c}\right)}{as^2+bs+c}ds$ ? Main issue here is the term inside $\sin^2()$. Very ...
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### Integral $\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$

I decided to follow a recent trend and ask a question about logarithmic integrals :) Is there a closed form for this integral? $$\int_0^1\frac{\log(x)\log^2(1-x)\log^2(1+x)}{x}\mathrm dx$$
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### Find the exact length of the curve $y=\frac 12 x^2- \frac 12 \ln(x)$

Find the exact length of the curve $y = \frac 12 x^2- \frac 12 \ln(x)$, for $2 \le x \le 4$. My attempt: \begin{align} L&= \int_2^4 \sqrt{1+\left[x-\frac 1{2x} \right]^2} \, dx \\ &= ...