Questions: [ int sqrt1-x d x ]

Solution Steps

We start with the integral \(\int \sqrt{1-x} \, dx\). To simplify this integral, we use the substitution \(u = 1 - x\), which gives us \(du = -dx\). Thus, the integral becomes \(-\int \sqrt{u} \, du\).

Next, we integrate \(-\int \sqrt{u} \, du\). The integral of \(\sqrt{u}\) is \(\frac{2}{3} u^{3/2}\). Therefore, we have: \[ -\int \sqrt{u} \, du = -\frac{2}{3} u^{3/2} + C \]

Now, we substitute back \(u = 1 - x\) into our result: \[ -\frac{2}{3} (1 - x)^{3/2} + C \]

Final Answer