Evaluate the integrals in Exercises 37–46.

∫(2θ + 1 + 2 cos (2θ + 1))dθ

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Identify the integral to be evaluated: \(\int (2\theta + 1 + 2 \cos(2\theta + 1)) \, d\theta\).

Split the integral into separate integrals for easier handling: \(\int (2\theta + 1) \, d\theta + \int 2 \cos(2\theta + 1) \, d\theta\).

Integrate the polynomial part \(\int (2\theta + 1) \, d\theta\) by applying the power rule: \(\int 2\theta \, d\theta\) and \(\int 1 \, d\theta\) separately.

For the trigonometric part \(\int 2 \cos(2\theta + 1) \, d\theta\), use substitution: let \(u = 2\theta + 1\), then find \(du\) and rewrite the integral in terms of \(u\).

Integrate \(\int 2 \cos(u) \cdot \frac{du}{2} = \int \cos(u) \, du\), then substitute back \(u = 2\theta + 1\) to express the result in terms of \(\theta\).

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주요 개념

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

Integration of Basic Functions

Integration involves finding the antiderivative of a function. For polynomials like 2θ + 1, integrate each term separately by increasing the power by one and dividing by the new exponent. Constants integrate to the constant times the variable.

Integration of Trigonometric Functions

To integrate trigonometric functions like cos(kθ + c), use substitution if necessary. The integral of cos(u) with respect to u is sin(u), so adjust for the inner function's derivative when integrating.

Use of Substitution Method

Substitution simplifies integration when the integrand contains a composite function. By setting u equal to the inner function (e.g., u = 2θ + 1), you rewrite the integral in terms of u, making it easier to integrate.

Evaluate the integrals in Exercises 37–46.∫(2θ + 1 + 2 cos (2θ + 1))dθ