Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
d. Using the substitution u = tan(x) in ∫ (tan²x / (tan x - 1)) dx leads to ∫ (u² / (u - 1)) du.

검증된 단계별 안내

Step 1: Begin by analyzing the substitution u = tan(x). When using substitution in integration, we replace the original variable (x) with a new variable (u) and also compute the derivative of the substitution. For u = tan(x), the derivative is du/dx = sec²(x), or equivalently, dx = du / sec²(x).

Step 2: Rewrite the original integral ∫ (tan²(x) / (tan(x) - 1)) dx in terms of u. Since tan(x) = u, tan²(x) becomes u², and tan(x) - 1 becomes u - 1. Substitute these expressions into the integral.

Step 3: Replace dx with du / sec²(x) in the integral. The integral now becomes ∫ (u² / (u - 1)) * (1 / sec²(x)) du. At this point, sec²(x) must also be expressed in terms of u to complete the substitution.

Step 4: Recall the trigonometric identity sec²(x) = 1 + tan²(x). Since tan(x) = u, sec²(x) = 1 + u². Substitute this into the integral, replacing sec²(x) with 1 + u².

Step 5: Simplify the integral. After substituting sec²(x) = 1 + u², the integral becomes ∫ (u² / (u - 1)) * (1 / (1 + u²)) du. This does not simplify directly to ∫ (u² / (u - 1)) du, so the statement in the problem is false. The substitution leads to a more complex integral.

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주요 개념

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

Substitution in Integration

Substitution is a technique used in integration to simplify the integrand by changing variables. By letting u = g(x), the integral can be transformed into a function of u, making it easier to evaluate. The differential dx is also converted using the derivative of g(x), which is essential for maintaining the equality of the integral.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the variables. In this context, knowing the identity tan²(x) = sec²(x) - 1 can help simplify the integrand before applying substitution. Understanding these identities is crucial for manipulating expressions involving trigonometric functions effectively.

Limits of Integration

When performing substitution in definite integrals, it is important to change the limits of integration to correspond with the new variable. This ensures that the area under the curve is accurately represented. In indefinite integrals, while limits are not a concern, understanding how they change with substitution is vital for definite integrals.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.d. Using the substitution u = tan(x) in ∫ (tan²x / (tan x - 1)) dx leads to ∫ (u² / (u - 1)) du.