Question

Evaluate the integral ∫(4x² sin(x)) dx.

Answer 1

To evaluate the integral ∫(4x² sin(x)) dx, we can use integration by parts. By assuming appropriate values for u and dv and using the integration by parts formula, we can find the integral value as -4x² cos(x) + 8x sin(x) + 8cos(x) + C.

Step-by-step explanation:

To evaluate the integral ∫(4x² sin(x)) dx, we can use integration by parts. Let's assume u = 4x² and dv = sin(x) dx. Taking the derivatives and integrals, we have du = 8x dx and v = -cos(x). Using the integration by parts formula ∫u dv = uv - ∫v du, we can evaluate the integral as follows:

1. ∫(4x² sin(x)) dx
2. Let u = 4x² and dv = sin(x) dx
3. du = 8x dx and v = -cos(x)
4. ∫(4x² sin(x)) dx = (-4x² cos(x)) - ∫(-cos(x) * 8x) dx
5. Simplifying, we get ∫(4x² sin(x)) dx = -4x² cos(x) + 8∫(x cos(x)) dx
6. The term 8∫(x cos(x)) dx can be evaluated using integration by parts again, but this time setting u = x and dv = cos(x) dx
7. Following the same steps as before, we find ∫(x cos(x)) dx = x sin(x) + ∫sin(x) dx
8. ∫(4x² sin(x)) dx = -4x² cos(x) + 8(x sin(x) - ∫sin(x) dx)
9. Simplifying further, we have ∫(4x² sin(x)) dx = -4x² cos(x) + 8x sin(x) - 8∫sin(x) dx
10. The integral ∫sin(x) dx is equal to -cos(x)
11. Substituting this back into the equation, we get ∫(4x² sin(x)) dx = -4x² cos(x) + 8x sin(x) + 8cos(x) + C, where C is the constant of integration

Therefore, the value of the integral ∫(4x² sin(x)) dx is -4x² cos(x) + 8x sin(x) + 8cos(x) + C, where C is the constant of integration.