integrate sin inverse x

Steps to Solve

1. Identify the Function to Integrate

We need to find the integral of the function $\sin^{-1}(x)$.

2. Use Integration by Parts

To solve the integral, we can use integration by parts, which is given by the formula: $$ \int u , dv = uv - \int v , du $$ Let's choose:

• $u = \sin^{-1}(x)$, then $du = \frac{1}{\sqrt{1 - x^2}} , dx$
• $dv = dx$, then $v = x$

3. Apply Integration by Parts

Now, applying the integration by parts formula: $$ \int \sin^{-1}(x) , dx = x \sin^{-1}(x) - \int x \frac{1}{\sqrt{1 - x^2}} , dx $$

4. Integrate the Remaining Integral

The integral $\int x \frac{1}{\sqrt{1 - x^2}} , dx$ can be solved by substitution. Let:

• $w = 1 - x^2$, then $dw = -2x , dx$ or $dx = -\frac{dw}{2x}$

Now substitute $w$ into the integral: $$ \int x \frac{1}{\sqrt{1 - x^2}} , dx = -\frac{1}{2} \int \frac{1}{\sqrt{w}} , dw $$

5. Solve the Remaining Integral

The integral of $\frac{1}{\sqrt{w}}$ is: $$ -\frac{1}{2} \cdot 2\sqrt{w} + C = -\sqrt{1 - x^2} + C $$

6. Combine All Parts Together

Now, substitute the result back into our equation: $$ \int \sin^{-1}(x) , dx = x \sin^{-1}(x) + \sqrt{1 - x^2} + C $$