Key Concepts and Formulas

• Integration by Parts: $\int u \, dv = uv - \int v \, du$
• Substitution Method: If $u = g(x)$, then $\int f(g(x))g'(x) \, dx = \int f(u) \, du$
• Given: $\int f(x) \, dx = \psi(x)$

Step-by-Step Solution

Step 1: Set up the integral and apply substitution

We are given the integral $I = \int x^5 f(x^3) \, dx$. We want to simplify this using a substitution. Let $t = x^3$. Then, $\frac{dt}{dx} = 3x^2$, which implies $dt = 3x^2 \, dx$. We rewrite the integral to prepare for the substitution:

$I = \int x^5 f(x^3) \, dx = \int x^3 \cdot x^2 f(x^3) \, dx = \int x^3 f(x^3) x^2 \, dx$

Now, we substitute $t = x^3$, so $x^3 = t$ and $x^2 \, dx = \frac{1}{3} \, dt$. The integral becomes:

$I = \int t f(t) \frac{1}{3} \, dt = \frac{1}{3} \int t f(t) \, dt$

Step 2: Apply integration by parts

We will use integration by parts on the integral $\int t f(t) \, dt$. Let $u = t$ and $dv = f(t) \, dt$. Then, $du = dt$ and $v = \int f(t) \, dt = \psi(t)$. Applying integration by parts:

$\int t f(t) \, dt = t \psi(t) - \int \psi(t) \, dt$

Step 3: Substitute back into the expression for I

Substitute this result back into the expression for $I$:

$I = \frac{1}{3} \left[ t \psi(t) - \int \psi(t) \, dt \right]$

Now substitute $t = x^3$ back into the expression:

$I = \frac{1}{3} \left[ x^3 \psi(x^3) - \int \psi(x^3) \, dt \right]$

Since $t=x^3$, we have $dt = 3x^2 dx$, so $dx = \frac{dt}{3x^2}$. Thus, $\int \psi(x^3) \, dt = \int \psi(x^3) 3x^2 dx$.

Therefore, $I = \frac{1}{3} \left[ x^3 \psi(x^3) - \int \psi(x^3) 3x^2 dx \right]$ $I = \frac{1}{3} \left[ x^3 \psi(x^3) - 3\int x^2 \psi(x^3) dx \right] + C$ $I = \frac{1}{3} \left[ x^3 \psi(x^3) \right] - \int x^2 \psi(x^3) dx + C$

$I = \frac{1}{3} x^3 \psi(x^3) - \int x^2 \psi(x^3) dx + C$

Step 4: Manipulate to match the correct option

We can rewrite the expression as:

$I = \frac{1}{3} \left[ x^3 \psi(x^3) - 3\int x^2 \psi(x^3) dx \right] + C$

$I = \frac{1}{3}\left[ {{x^3}\psi \left( {{x^3}} \right) - 3\int {{x^2}\psi \left( {{x^3}} \right)dx} } \right] + C$

Common Mistakes & Tips

• Remember to substitute back to the original variable after integration by parts or substitution.
• Be careful with the constants when applying the substitution method. Specifically, make sure to account for the $3x^2$ term when substituting $t = x^3$.
• Double-check your integration by parts setup, especially the choice of $u$ and $dv$.

Summary

We started with the given integral and used a combination of substitution and integration by parts to evaluate it. First, we substituted $t = x^3$ to simplify the integral. Then, we applied integration by parts and finally, we substituted back to the original variable to obtain the final result. The final answer is $\frac{1}{3}\left[ {{x^3}\psi \left( {{x^3}} \right) - 3\int {{x^2}\psi \left( {{x^3}} \right)dx} } \right] + C$.

Final Answer The final answer is $\boxed{{1 \over 3}\left[ {{x^3}\psi \left( {{x^3}} \right) - 3\int {{x^2}\psi \left( {{x^3}} \right)dx} } \right] + C}$, which corresponds to option (A).