Evaluate the integrals in Exercises 47–68.
_
∫₁⁴ (1 + √u)¹/² du
√u

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First, rewrite the integral to clarify the expression. The integral is \( \int_1^4 (1 + \sqrt{u})^{1/2} \, du \). Here, \( (1 + \sqrt{u})^{1/2} \) means the square root of \( 1 + \sqrt{u} \).

To simplify the integral, use a substitution. Let \( t = \sqrt{u} \), which means \( t = u^{1/2} \). Then, express \( du \) in terms of \( dt \). Since \( u = t^2 \), differentiate both sides to get \( du = 2t \, dt \).

Change the limits of integration to match the substitution. When \( u = 1 \), \( t = \sqrt{1} = 1 \). When \( u = 4 \), \( t = \sqrt{4} = 2 \). So the new limits for \( t \) are from 1 to 2.

Rewrite the integral in terms of \( t \): \( \int_1^2 (1 + t)^{1/2} \cdot 2t \, dt \). This simplifies to \( 2 \int_1^2 t (1 + t)^{1/2} \, dt \).

Now, to evaluate \( 2 \int_1^2 t (1 + t)^{1/2} \, dt \), consider using integration by parts or another substitution such as \( w = 1 + t \) to simplify the integral further.

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주요 개념

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

Definite Integrals

A definite integral calculates the net area under a curve between two specific limits. It is represented as ∫_a^b f(x) dx, where a and b are the lower and upper bounds. Evaluating definite integrals involves finding the antiderivative and then applying the Fundamental Theorem of Calculus.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It involves setting a new variable equal to a function inside the integral, then rewriting the integral in terms of this variable and its differential.

Handling Radicals in Integrals

Integrals involving radicals, such as square roots, often require rewriting the expression using fractional exponents. This allows the use of power rule integration techniques. Simplifying the integrand before integrating makes the process more straightforward.

Evaluate the integrals in Exercises 47–68. _∫₁⁴ (1 + √u)¹/² du √u