So, I have the following indefinite integral:  $$\int sin(u)e^u \ du$$  I tried solving it using integration by parts, but it just kept repeating itself. What should I do?
81178
I'm trying to solve the integral of $\left(\int e^x\,\sin x\,\,dx\right)$ (My solution):  $\int e^x\sin\left(x\right)\,\,dx=$  $\int \sin\left(x\right) \,e^x\,\,dx=$  $\left(\sin(x)\,\int e^x\right)-\left(\int\sin^{'}(x)\,\left(\int e^x\right)\right)$  $\left(\sin(x)\,e^x\right)-\left(\int\cos(x)\,e^x\right)$  $\left(\sin(x)\,e^x\right)-\left(\cos(x)\,e^x-\left(\int-\sin\left(x\right)\,e^x\right)\right)$  $\left(\sin(x)\,e^x\right)-\left(\cos(x)\,e^x-\left(-\sin\left(x\right)\,e^x-\int-\cos\left(x\right)\,e^x\right)\right)$  I don't know how to complete because the solution gonna be very complicated.
32634
Indefinite Integral with sin/cos  I can't find a good way to integrate: $$\int\dfrac{3\sin(x) + 2\cos(x)}{2\sin(x) + 3\cos(x)} \; dx $$
87616
In evaluating integral $\int\sin^{3}x dx $ I am pretty sure we need to use substitution $e^x=t$, but can't go  next step.
71393
Find a general formula for evaluating integrals of the form $\displaystyle\int_0^\infty x^a e^{-bx}dx.$   I've been pondering about this question for some time. Obviously integration by parts is way too inefficient. I know there's recursion involved and maybe there's a way to find a general formula through intuition.
81210
As the title states, I'm having quite a lot of trouble with the integral:  $$\int_{-\ln 2}^{\ln 2}  \frac{(\sin x+x)^{1/3}}{e^x}dx$$  The problem is that my standard (admittedly sparse) repertoire of tricks seems to have no effect at all. I can't think of any clever substitution that would work, and it looks like a nasty integral all-round.   I was hoping to apply some symmetry arguments (especially considering the limits of integration), but the exponential function is neither even nor odd, so I don't see how this would work.   I'm afraid I have absolutely no idea how to approach this.
114276
Integrate $$\int \sin^4x \cos^2x dx$$  Now, there's few solutions to this problem already on the internet. For example on yahoo:  and MIT   I've tried a slightly different approach and simply wonder if my answer is correct as I don't know how to check it.  \begin{eqnarray} \int \sin^4x \cos^2x dx &amp;=&amp; \int \left( \frac{1-\cos^2x}{2} \right)^2 \cos^2x dx \\ &amp;=&amp; \frac{1}{4} \int (1-2\cos^2x+\cos^4x)\cos^2x dx \\ &amp;=&amp;  \frac{1}{4} \int \cos^6x-2\cos^4x+\cos^2xdx \\ &amp;=&amp;  \frac{1}{4} \left( \frac{1}{7}\cos^7x-\frac{2}{5}\cos^5x+\frac{1}{3}\cos^3x \right) + C \end{eqnarray}
88098
$\int e^{\frac{x^2}{2}} \cdot x^{3} dx=?$  I tried to do the substitution $du = x^3dx$, so that $u=\frac{x^4}{4}$. Then, $\int e^{\frac{x^2}{2}} \cdot x^{3} dx= \int e^{\sqrt{u}} du$  However, I would have to do a second substitution ($v = \sqrt{u}$) which would give a product again.  How should I proceed?
17869
$$\int \frac{\sin\left(x\right)}{1+x^2}dx$$  I have tried to integrate by parts but it doesn't work. How do I evaluate it? Any advice, hint or well-thought solution will be appreciated.
57087
I came across a list of functions in my calculus textbook whose indefinite integral cannot be found. It was written that the integral $$\int \frac{\sin(x)}{x} dx$$ cannot be evaluated without any explanation as to why.  I did some research over the internet and found out that the definite integral $$\int_{0}^{\infty} \frac{\sin(x)}{x} dx$$ can be evaluated using Laplace transformation and is equal to $\pi /2$. But I still couldn't find answer to my original question. I read somewhere that the integral cannot be expressed using 'elementary functions'. A little help is appreciated, I'm in Calc 1 going advanced than my course but I am sorry if my post shows lack of research. Thank you!
98396
Can you please provide any sort of hint or suggestion on how to find the following indefinite integral?  $$\int\frac{x^3}{\sqrt{x^2+1}}\text{d}x$$  I tried substituting everything but it didn't work. I also tried trigonometric substitution but I couldn't find any valid trigonometric identity for $ \sqrt{\cos^2(x)+1} $ or $ \sqrt{\sin^2(x)+1} $.
120211
I need help calculation the integral $\int_{-\infty}^{\infty} x\sin(x)/(x^2+1)^2 dx$  I am getting confused on how to integrate around $\pm i$. I think I'll have to use $\gamma = Re^{i\theta}$ where $|R|\leq n$ for $n\geq1$
118635
How can I solve this indefinite integral for an arbitrary integer $n&gt;0$?  $$ \int{x^n e^x dx}$$  I could partially integrate it for small $n$, but that's not really a solution.    Edit: (TB) This question is closely related to: , but it is more elementary, because $n$ is an integer here.
75214
How do I integrate the following $$\int \frac{dx}{\sin^3 x + \cos^3 x}$$ ?  It appears that I am supposed to break this up into $(\sin x + \cos x)(1-\cos x \sin x)$, but the next thing to do is not apparent to me.
98479
I've encountered this integral many, many times (exams, exercises) and always end up wasting a bit of time calculating it for different $\rm a,b$'s.  Is there any way to calculate the following integral?  $$\rm \int x^ae^{-bx}dx.\quad  a,b \in \Bbb Z_{\geq 0}.$$
69250
$\int_{0}^{1}\frac{\sin^{-1}\sqrt x}{x^2-x+1}dx$  Put $x=\sin^2\theta,dx=2\sin \theta \cos \theta d\theta$  $\int_{0}^{\pi/2}\frac{\theta.2\sin \theta \cos \theta d\theta}{\sin^4\theta-\sin^2\theta+1}$ but this seems not integrable.Is this a wrong way?Is there a different way to solve it?
116269
$$\int_0^1\sqrt\frac x{1-x}\,dx$$ I saw in my book that the solution is $x=\cos^2u$ and $dx=-2\cos u\sin u\ du$. I would like to see different approaches, can you provide them?
64120
I recently came across this problem:  Prove that $\int_{0}^{\frac{\pi}{2}} \sin(\sin y) dy \leq 1$.  However, from my understanding, the expression $\sin(\sin y)$ does not have an indefinite integral (). So, how will we go about solving the above problem using just the Riemann sums?