1 / | | / 2 2\ | log\x + y / dy | / 0
Integral(log(x^2 + y^2), (y, 0, 1))
Use integration by parts:
Let and let .
Then .
To find :
The integral of a constant is the constant times the variable of integration:
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:
Rewrite the integrand:
Integrate term-by-term:
The integral of a constant times a function is the constant times the integral of the function:
The integral of is .
So, the result is:
The integral of a constant is the constant times the variable of integration:
The result is:
So, the result is:
Add the constant of integration:
The answer is:
2 / y \ 2*x *atan|-------| / | ____| | | / 2 | | / 2 2\ / 2 2\ \\/ x / | log\x + y / dy = C - 2*y + y*log\x + y / + ------------------ | ____ / / 2 \/ x
/I*log(1 - I*x) I*log(1 + I*x)\ /I*log(-I*x) I*log(I*x)\ / 2\ -2 - 2*x*|-------------- - --------------| + 2*x*|----------- - ----------| + log\1 + x / \ 2 2 / \ 2 2 /
=
/I*log(1 - I*x) I*log(1 + I*x)\ /I*log(-I*x) I*log(I*x)\ / 2\ -2 - 2*x*|-------------- - --------------| + 2*x*|----------- - ----------| + log\1 + x / \ 2 2 / \ 2 2 /
-2 - 2*x*(i*log(1 - i*x)/2 - i*log(1 + i*x)/2) + 2*x*(i*log(-i*x)/2 - i*log(i*x)/2) + log(1 + x^2)
Use the examples entering the upper and lower limits of integration.