# Tagged Questions

Questions about the evaluation of specific definite integrals.

50 views

### Is it always possible to find a $t>0$, such that $\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|dx<C~~~?$

Is it always possible to find a $t>0$, such that $$\int_{0}^{t}|\sum_{k=1}^{n}\cos kx|\,dx<C~~~?$$ where $C$ is independent of $n$. Here is my idea: We know that \begin{align} ...
90 views

### Without performing the integral to show

Without performing the integral, show that $$\int_{-2}^{2}e^{x^2}x^3dx=0$$ Please help me. I do not know where to start.
66 views

### How does one prove this formula?

Show that $$\int_0^{\infty}\frac{1}{\sqrt{x}}\exp\left[-a^2 x\left(\frac{x-6}{x-2}\right)^2\right]\,dx=\frac{\sqrt{\pi}}{a},$$ where $a$ is real.
248 views

### Integral $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{x\sin{x}}{1+\cos^4{x}}dx$

Question: Find the integral $$I=\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\dfrac{x\sin{x}}{1+\cos^4{x}}dx$$ my try: since $$I=2\int_{0}^{\frac{\pi}{2}}\dfrac{x\sin{x}}{1+\cos^4{x}}dx$$ then I can't. I ...
31 views