6. Evaluate ∫ cos x √(100 − sin² x) dx using tables after performing the substitution u = sin x.

검증된 단계별 안내

Recognize that the integral involves a trigonometric function and a square root expression. The substitution u = sin(x) is suggested to simplify the integral. Start by calculating the derivative of u: du = cos(x) dx.

Substitute u = sin(x) into the integral. Replace sin²(x) with u² and cos(x) dx with du. The integral becomes ∫ √(100 − u²) du.

Notice that the integral ∫ √(a² − u²) du matches a standard form in integral tables. Specifically, it corresponds to the formula for ∫ √(a² − u²) du = (1/2) [u √(a² − u²) + a² arcsin(u/a)] + C, where a is a constant.

Identify the constant a in the given integral. Here, a² = 100, so a = 10. Substitute a = 10 into the standard formula.

After applying the formula, back-substitute u = sin(x) to express the result in terms of x. This completes the evaluation of the integral.

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주요 개념

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

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots of expressions containing trigonometric functions. By substituting a trigonometric function for a variable, we can transform the integral into a more manageable form. In this case, substituting u = sin x allows us to express the integral in terms of u, simplifying the evaluation process.

Integration by Substitution

Integration by substitution is a method that simplifies the process of finding integrals by changing the variable of integration. This technique involves substituting a new variable, which often makes the integral easier to evaluate. In the given question, the substitution u = sin x will help in transforming the integral into a form that can be evaluated using standard integral tables.

Integral Tables

Integral tables are collections of pre-calculated integrals that provide quick references for evaluating common integrals. They are particularly useful when dealing with complex functions that do not have straightforward antiderivatives. After performing the substitution in the given integral, one can refer to these tables to find the integral of the resulting expression, facilitating a faster solution.