Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ arcsin(y) dy

Guida verificata passo dopo passo

Identify the integral to solve: $\int \arcsin(y) \, dy$.

Recall the integration by parts formula: $\int u \, dv = uv - \int v \, du$.

Choose $u = \arcsin(y)$ because its derivative simplifies, and $dv = dy$ because it is easy to integrate.

Compute $du$ and $v$: - $du = \frac{1}{\sqrt{1 - y^2}} \, dy$ (derivative of $\arcsin(y)$), - $v = y$ (integral of $dy$).

Apply the integration by parts formula: $\int \arcsin(y) \, dy = y \arcsin(y) - \int y \cdot \frac{1}{\sqrt{1 - y^2}} \, dy$. Next, focus on evaluating the remaining integral $\int \frac{y}{\sqrt{1 - y^2}} \, dy$.

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