Use u-substitution to find the indefinite integral
β« 10π§π^βπ§^2ππ§ . Please clearly show all
work.
Accepted Solution
A:
To find the indefinite integral of β« 10π§π^βπ§^2ππ§ using u-substitution, we can let u = -z^2. Let's go through the steps:
Let u = -z^2. Differentiating both sides with respect to z, we get du = -2z dz, which implies dz = -(1/2z) du.
Substitute u and dz in terms of du in the integral:
β« 10π§π^βπ§^2ππ§ = β« 10z e^u dz = β« 10z e^u (-1/2z) du.
Simplify the expression:
The z in the numerator and denominator cancel out, leaving us with:
β« -5e^u du.
Integrate with respect to u:
β« -5e^u du = -5 β« e^u du = -5e^u + C,
where C is the constant of integration.
Substitute back u = -z^2:
-5e^u + C = -5e^-z^2 + C.
Therefore, the indefinite integral of β« 10π§π^βπ§^2ππ§ is -5e^-z^2 + C.