is the indefinite integral of 4x^3sinx^4dx = -cosx^4+c
To determine if the indefinite integral of 4x^3sinx^4dx is indeed -cosx^4+c, we can use the power rule for integration and make a substitution.
Step 1: Perform a substitution:
Let u = x^4. Then, du = 4x^3dx.
Step 2: Rewrite the integral:
The integral becomes ∫ sin(u)du.
Step 3: Integrate:
Integrating sin(u)du gives us -cos(u) + C, where C is the constant of integration.
Step 4: Substitute back the original variable:
Replace u with x^4 to get -cos(x^4) + C.
Therefore, the indefinite integral of 4x^3sinx^4dx is indeed -cosx^4 + C, where C is the constant of integration.