Question

Evaluate the integral using integration by parts. ∫3sin⁻¹(x) dx?

Answer 1

To evaluate the integral 3sin⁻¹(x) dx using integration by parts, we can choose u = sin⁻¹(x) and dv = 3 dx. By applying the integration by parts formula, we can simplify the integral and potentially reduce it to another integral.

Step-by-step explanation:

To evaluate the integral ∫3sin⁻¹(x) dx using integration by parts, we need to apply the integration by parts formula:

∫ u dv = uv - ∫ v du

We can choose:

u = sin⁻¹(x) and dv = 3 dx

taking the derivatives and integrals:

du = 1/√(1-x²) dx and v = 3x

Substituting these values into the formula, we get:

∫3sin⁻¹(x) dx = 3(xsin⁻¹(x)) - ∫ 3x(1/√(1-x²)) dx

Simplifying further, we get:

∫3sin⁻¹(x) dx = 3(xsin⁻¹(x)) + 3∫(x/√(1-x²)) dx

To evaluate the remaining integral, we can use a trigonometric substitution or integration by parts again.