1 / | | / 2\ | log\1 + x / dx | / 0
Integral(log(1 + x^2), (x, 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 is the constant times the variable of integration:
The integral of a constant times a function is the constant times the integral of the function:
PiecewiseRule(subfunctions=[(ArctanRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), True), (ArccothRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False), (ArctanhRule(a=1, b=1, c=1, context=1/(x**2 + 1), symbol=x), False)], context=1/(x**2 + 1), symbol=x)
So, the result is:
The result is:
So, the result is:
Add the constant of integration:
The answer is:
/ | | / 2\ / 2\ | log\1 + x / dx = C - 2*x + 2*atan(x) + x*log\1 + x / | /
pi -2 + -- + log(2) 2
=
pi -2 + -- + log(2) 2
-2 + pi/2 + log(2)
Use the examples entering the upper and lower limits of integration.