Evaluate the integrals in Exercises 47–68.
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∫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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