Help me with some short integrals!

[OrangeOne]

Hello!

I am having problem solving these ones:


? -y/(x^2+y^2)dx

?t* sqrt(1+4t^2) dt


I haven't done anything so far because I don't know where to start

I also can't solve g(x) in the following equation:

(g'(x)-g(x))x sin xy = 0

 

[galactus]

I am having problem solving these ones:

\(\displaystyle \int\frac{-y}{x^{2}+y^{2}}dx\)

This is not a double integral, so I reckon y is a constant.

\(\displaystyle \int\frac{-y}{x^{2}+y^{2}}dx\)

You can use a trig sub. One of various ways, but about as good as any.

Let \(\displaystyle x=y\cdot tan{\theta}, \;\ dx=y\cdot sec^{2}{\theta}d{\theta}\)

\(\displaystyle \int\frac{-y}{(y\cdot tan{\theta})^{2}+y^{2}}\cdot y\cdot sec^{2}{\theta}d{\theta}\)

\(\displaystyle \int\frac{-y\cdot y\cdot sec^{2}{\theta}}{y^{2}(tan^{2}{\theta}+1)}d{\theta}\)

Use the Identity \(\displaystyle tan^{2}{\theta}+1=sec^{2}{\theta}\)

It whittles down nicely. Finish?.

\(\displaystyle \int t\sqrt{1+4t^{2}}dt\)

Let \(\displaystyle u=1+4t^{2}, \;\ du=8tdt\Rightarrow \frac{du}{8}=tdt\)

 

[Deleted member 4993]

Hello!

I am having problem solving these ones:


? -y/(x^2+y^2)dx

?t* sqrt(1+4t^2) dt


I haven't done anything so far because I don't know where to start

I also can't solve g(x) in the following equation:

(g'(x)-g(x))x sin xy = 0

Start with

g'(x) = g(x)

g'(x)/g(x) = 1

Now integrate both sides and continue....