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Chapter 4: Problem 71

Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{0}^{4} \frac{1}{\sqrt{2 x+1}} d x $$

Short Answer

Expert verified
The definite integral evaluates to \(4\sqrt{2}-2\)

Step by step solution

01

Recognize the Integral Form

Recognize that the given integral is in the form of \[ \int \frac{1}{\sqrt{ax+b}} dx\] where \(a = 2\) and \(b = 1\). This form suggests the use of substitution method in integral calculus.
02

Apply substitution

The substitution method involves setting \(u = 2x + 1\). Differentiating both sides of the equation with respect to \(x\), we get \(du = 2dx\) or \(dx = du / 2\). Substituting the values obtained from this transformation into the original expression, we have \[ \int \frac{1}{\sqrt{u}}\cdot \frac{1}{2} du\] The integral limits change accordingly from \(x = 0\) to \(u = 1\) and from \(x = 4\) to \(u = 9\) yielding \[ \frac{1}{2}\int_{1}^{9} u^{-1/2} du \]
03

Solve the Transformed Integral

Now we can solve the transformed integral. It falls under the power rule which states that \[\int x^n dx = \frac{x^{n+1}}{n+1}\] Upon applying this rule, we get \[ \frac{u^{1/2}}{1/2}\] between the limits \(1\) and \(9\). That simplifies to \(2u^{1/2}\). Using the limits, we evaluate this to get: \(2*(9^{1/2}) - 2*(1^{1/2}) = 4\sqrt{2} - 2\]. Remember to replace 'u' with '2x+1' to revert to the original variable: \[2*(2*4 + 1)^{1/2} - 2*(2*0 + 1)^{1/2} = 4\sqrt{2}-2\].
04

Conclusion and final answer

Through the algebraic process of substitution and using the power rule in integration, the definite integral evaluates to \(4\sqrt{2}-2\) as the final answer.

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