Complete the integrals
$\displaystyle\int \frac{\sin(x)dx}{\sin(x)+\cos(x)}$
$\displaystyle\int_{0}^{\infty } x^{-5}\sin(x)dx$
Find the following limit
$\displaystyle\lim_{x \to \infty } \frac{x^{3}+\sin^{3}x}{x^{3}+\cos^{3}x}$
Thanks!
Complete the integrals
$\displaystyle\int \frac{\sin(x)dx}{\sin(x)+\cos(x)}$
$\displaystyle\int_{0}^{\infty } x^{-5}\sin(x)dx$
Find the following limit
$\displaystyle\lim_{x \to \infty } \frac{x^{3}+\sin^{3}x}{x^{3}+\cos^{3}x}$
Thanks!
HINTS:
For (1): Multiply the integrand by $\dfrac{\sin x-\cos x}{\sin x-\cos x}$ and use the double and half angle formulas to get a fraction containing only sines and cosines of $2x$ instead of $x$. If you do it right, the denominator will be a single trig function of $2x$, so you’ll be able to divide it through to get an integrand with no fractions.
For (2): I don’t see a straightforward integration, but the following indirect argument will work. Let $f(x)=x^{-5}\sin x$. Then $f(x)<0$ only when $(2n+1)\pi
On the other hand, there is a positive $a<\pi/2$ such that if $0
$\int_b^a\frac{\sin x}{x^5}dx\ge\frac12\int_b^ax^{-4}dx=-\frac16\left[x^{-3}\right]_b^a=\frac16\left[x^{-3}\right]_a^b=\frac16\left(\frac1{b^3}-\frac1{a^3}\right)\;.$ What is the limit of this as $b\to 0^+$? What can you conclude about $\displaystyle\int_0^\infty f(x)dx$?
For (3): This is very straightforward. How would you handle $\lim_{x\to\infty}\frac{x^3+1}{x^3-2}\;?$ There’s a standard technique, and it works equally well on this problem. Remember, $-1\le \sin x,\cos x\le 1$, and you have the squeeze theorem to work with.