1 / | | 15*x*atan(x) dx | / 0
Integral((15*x)*atan(x), (x, 0, 1))
Use integration by parts:
Let and let .
Then .
To find :
The integral of a constant times a function is the constant times the integral of the function:
The integral of is when :
So, the result is:
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 | 15*x 15*atan(x) 15*x *atan(x) | 15*x*atan(x) dx = C - ---- + ---------- + ------------- | 2 2 2 /
15 15*pi - -- + ----- 2 4
=
15 15*pi - -- + ----- 2 4
-15/2 + 15*pi/4
Use the examples entering the upper and lower limits of integration.