Question

What are the guidelines on integration by parts?

Answer 1

I know that:
when you have (for example) :
\[\int\limits_{ }^{ }\ln(x)~dx\]
then your another function for by parts, is (will give it a notation,) g(x)=1.
And you can not choose to differentiate g(x).

Answer 2

I am asking, would I be allowed to?
\[\int\limits_{ }^{ }x~dx=x^2-\int\limits_{ }^{ }x~dx\]\[\int\limits_{ }^{ }x~dx=x^2-x^2+\int\limits_{ }^{ }x~dx\]\[\int\limits_{ }^{ }x~dx=x^2-x^2+x^2-\int\limits_{ }^{ }x~dx\](if I am doing that correctly)

Answer 3

\[\int\limits_{}^{}x dx=\int\limits_{}^{}1 x dx=x(x)-\int\limits_{}^{}x(1) dx \\ \text{ add on both sides the } \int\limits_{}^{}x dx \\ 2 \int\limits_{}^{}x dx=x(x) \\ \int\limits_{}^{}x dx=\frac{1}{2}x(x) \\ \int\limits_{}^{}x dx=\frac{1}{2}x(x)+C\]

Answer 4

That could work for integrate x w.r.t x by integration by parts

Answer 5

yours seems to be going in a circle

Answer 6

\[n \neq -1 \\ \int\limits_{}^{}x^n dx=\int\limits_{}^{}1 x^{n} dx=x(x^n)-\int\limits_{}^{}x(n x^{n-1}) dx \\ \int\limits_{}^{}x^n dx=x(x^n)-n \int\limits_{}^{}x^{n} dx \\ \text{ add on both sides the } n \int\limits_{}^{}x^n dx \\ (n+1)\int\limits_{}^{}x^n dx=x(x^n) \\ \int\limits_{}^{}x^n dx=\frac{x(x^n)}{n+1}+C\]

Answer 7

like you had it you just needed to add the integral thing on both sides in your first step