$$ \int \sqrt { x } \sin ^ { 2 } ( x ^ { 3 / 2 } - 1 ) d x , \quad u = x ^ { 3 / 2 } - 1 $$

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Evaluate the integral by following steps:
- step0: Evaluate using substitution:
  $$\int \sqrt{x}\times \sin^{2}\left(x^{\frac{3}{2}}-1\right) dx$$
- step1: Rewrite the expression:
  $$\int x^{\frac{1}{2}}\sin^{2}\left(x^{\frac{3}{2}}-1\right) dx$$
- step2: Use the substitution $$dx=\frac{2}{3x^{\frac{1}{2}}} dt$$ to transform the integral$$:$$
  $$\int x^{\frac{1}{2}}\sin^{2}\left(x^{\frac{3}{2}}-1\right)\times \frac{2}{3x^{\frac{1}{2}}} dt$$
- step3: Simplify:
  $$\int \frac{2\sin^{2}\left(x^{\frac{3}{2}}-1\right)}{3} dt$$
- step4: Use the substitution $$t=x^{\frac{3}{2}}$$ to transform the integral$$:$$
  $$\int \frac{2\sin^{2}\left(t-1\right)}{3} dt$$
- step5: Simplify the expression:
  $$\int \frac{2}{3}\sin^{2}\left(t-1\right) dt$$
- step6: Use properties of integrals:
  $$\frac{2}{3}\times \int \sin^{2}\left(t-1\right) dt$$
- step7: Use the substitution $$dt=1 dv$$ to transform the integral$$:$$
  $$\frac{2}{3}\times \int \sin^{2}\left(t-1\right)\times 1 dv$$
- step8: Simplify:
  $$\frac{2}{3}\times \int \sin^{2}\left(t-1\right) dv$$
- step9: Use the substitution $$v=t-1$$ to transform the integral$$:$$
  $$\frac{2}{3}\times \int \sin^{2}\left(v\right) dv$$
- step10: Evaluate the integral:
  $$\frac{2}{3}\left(\frac{v}{2}-\frac{\sin\left(2v\right)}{4}\right)$$
- step11: Simplify the expression:
  $$\frac{2}{3}\left(\frac{v}{2}-\frac{1}{4}\sin\left(2v\right)\right)$$
- step12: Use the distributive property:
  $$\frac{2}{3}\times \frac{v}{2}+\frac{2}{3}\left(-\frac{1}{4}\sin\left(2v\right)\right)$$
- step13: Multiply the terms:
  $$\frac{v}{3}+\frac{2}{3}\left(-\frac{1}{4}\sin\left(2v\right)\right)$$
- step14: Multiply the terms:
  $$\frac{v}{3}-\frac{1}{6}\sin\left(2v\right)$$
- step15: Substitute back:
  $$\frac{t-1}{3}-\frac{1}{6}\sin\left(2\left(t-1\right)\right)$$
- step16: Multiply the terms:
  $$\frac{t-1}{3}-\frac{1}{6}\sin\left(2t-2\right)$$
- step17: Simplify the expression:
  $$\frac{1}{3}t-\frac{1}{3}-\frac{1}{6}\sin\left(2t-2\right)$$
- step18: Substitute back:
  $$\frac{1}{3}x^{\frac{3}{2}}-\frac{1}{3}-\frac{1}{6}\sin\left(2x^{\frac{3}{2}}-2\right)$$
- step19: Simplify:
  $$\frac{1}{3}x\sqrt{x}-\frac{1}{3}-\frac{1}{6}\sin\left(2x^{\frac{3}{2}}-2\right)$$
- step20: Calculate:
  $$\frac{1}{3}x\sqrt{x}-\frac{1}{3}-\frac{1}{6}\sin\left(2x\sqrt{x}-2\right)$$
- step21: Add the constant of integral C:
  $$\frac{1}{3}x\sqrt{x}-\frac{1}{3}-\frac{1}{6}\sin\left(2x\sqrt{x}-2\right) + C, C \in \mathbb{R}$$
To solve the integral

$$
\int \sqrt{x} \sin^2(x^{3/2} - 1) \, dx,
$$

we will use the substitution $$ u = x^{3/2} - 1 $$.

### Step 1: Find $$ du $$

First, we need to differentiate $$ u $$ with respect to $$ x $$:

$$
u = x^{3/2} - 1 \implies \frac{du}{dx} = \frac{3}{2} x^{1/2} \implies du = \frac{3}{2} x^{1/2} \, dx \implies dx = \frac{2}{3} \frac{du}{x^{1/2}}.
$$

### Step 2: Substitute in the integral

Now, we can substitute $$ u $$ and $$ dx $$ into the integral:

$$
\int \sqrt{x} \sin^2(u) \left(\frac{2}{3} \frac{du}{x^{1/2}}\right) = \frac{2}{3} \int \sin^2(u) \, du.
$$

### Step 3: Integrate $$ \sin^2(u) $$

Using the identity $$ \sin^2(u) = \frac{1 - \cos(2u)}{2} $$:

$$
\int \sin^2(u) \, du = \int \frac{1 - \cos(2u)}{2} \, du = \frac{1}{2} \left( u - \frac{1}{2} \sin(2u) \right) + C.
$$

### Step 4: Substitute back

Now substituting back for $$ u $$:

$$
\int \sin^2(u) \, du = \frac{1}{2} \left( x^{3/2} - 1 - \frac{1}{2} \sin(2(x^{3/2} - 1)) \right) + C.
$$

### Step 5: Combine everything

Putting it all together, we have:

$$
\frac{2}{3} \left( \frac{1}{2} \left( x^{3/2} - 1 - \frac{1}{2} \sin(2(x^{3/2} - 1)) \right) \right) + C.
$$

This simplifies to:

$$
\frac{1}{3} x \sqrt{x} - \frac{1}{3} - \frac{1}{6} \sin(2(x^{3/2} - 1)) + C.
$$

### Final Result

Thus, the integral evaluates to:

$$
\frac{1}{3} x \sqrt{x} - \frac{1}{3} - \frac{1}{6} \sin(2(x^{3/2} - 1)) + C, \quad C \in \mathbb{R}.
$$
The integral evaluates to: $$ \frac{1}{3} x \sqrt{x} - \frac{1}{3} - \frac{1}{6} \sin(2(x^{3/2} - 1)) + C $$ where $$ C $$ is the constant of integration.

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