Question

Evaluate: $\int x^{2}\sin(x^{3})\,dx$

Hint:

Always check if the derivative of the inner function exists in the integrand when using substitution.

Answer 1

Step 1: Substitution. Let $t = x^{3} \implies dt = 3x^{2}dx \implies x^{2}dx = \dfrac{dt}{3}$.

Step 2: Transform the integral. \[ \int x^{2}\sin(x^{3})\,dx = \int \sin(t)\,\dfrac{dt}{3} \]

Step 3: Solve. \[ = \dfrac{1}{3}\int \sin(t)\,dt = -\dfrac{1}{3}\cos(t) + C \]

Step 4: Back-substitution. \[ = -\dfrac{1}{3}\cos(x^{3}) + C \]

Final Answer: \[ \boxed{-\dfrac{1}{3}\cos(x^{3}) + C} \]