Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² sin(x³) dx

검증된 단계별 안내

Identify the integral to solve: $\int x^{2} \sin(x^{3}) \, dx$.

Look for a substitution that simplifies the integral. Notice that the argument of the sine function is $x^{3}$, and the derivative of $x^{3}$ is \$3x^{2}$, which is similar to the $x^{2}$ term outside the sine.

Set the substitution: let $u = x^{3}$. Then, compute the differential: $du = 3x^{2} \, dx$, which implies $x^{2} \, dx = \frac{du}{3}$.

Rewrite the integral in terms of $u$: $\int x^{2} \sin(x^{3}) \, dx = \int \sin(u) \cdot \frac{du}{3} = \frac{1}{3} \int \sin(u) \, du$.

Integrate $\sin(u)$ with respect to $u$: $\int \sin(u) \, du = -\cos(u) + C$. Then substitute back $u = x^{3}$ to express the answer in terms of $x$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. For example, setting u = x³ converts the integral into one involving sin(u), making it easier to integrate.

Integration by Parts

Integration by parts is a technique based on the product rule for differentiation, used to integrate products of functions. It is applied when substitution is not straightforward, but in this problem, it may not be necessary.

Recognizing When to Use Each Technique

Identifying the most efficient integration method is crucial. Here, recognizing that the integral can be solved by substitution rather than integration by parts saves time and simplifies the process.

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.∫ x² sin(x³) dx