Question

Evaluate the given integral. $$ \int_{3}^{4} \frac{1}{t-\sqrt{2 t}} d t $$

Short answer

The integral evaluates to \( 2\sqrt{2} \ln \left( \frac{2\sqrt{2} - 2}{\sqrt{6} - 2} \right) \).

Step-by-step solution

Analyze the Integral

First, recognize that the integral given is \( \int_{3}^{4} \frac{1}{t-\sqrt{2t}} \, dt \). The expression in the integrand is somewhat complex, and we should consider simplifying it by a substitution.

Perform Substitution

Notice the term \( \sqrt{2t} \). Let \( u = \sqrt{2t} \). This implies that \( u^2 = 2t \), or \( t = \frac{u^2}{2} \). Differentiating both sides with respect to \( t \), we have \( dt = \frac{u}{\sqrt{2}} du \).Also, adjust the bounds: when \( t = 3 \), \( u = \sqrt{6} \) and when \( t = 4 \), \( u = \sqrt{8} \). This changes the integral to:\[ \int_{\sqrt{6}}^{\sqrt{8}} \frac{\sqrt{2}}{\frac{u^2}{2} - u} \cdot u \, du \]

Simplify the New Integrand

Simplify the integrand \( \frac{\sqrt{2} \cdot u}{\frac{u^2}{2} - u} \). This becomes \( \frac{\sqrt{2} \cdot u}{\frac{1}{2}(u^2 - 2u)} = \frac{2\sqrt{2}\cdot u}{u(u - 2)} = \frac{2\sqrt{2}}{u - 2} \). The integral is now:\[ \int_{\sqrt{6}}^{\sqrt{8}} \frac{2\sqrt{2}}{u - 2} \, du \]

Integrate the Simplified Expression

The integral \( \int \frac{2\sqrt{2}}{u - 2} \, du \) can be easily solved using the natural logarithm. The antiderivative is:\[ 2\sqrt{2} \ln|u - 2| \]

Evaluate the Definite Integral

Substitute the bounds \( \sqrt{6} \) and \( \sqrt{8} \) into the antiderivative:\[ 2\sqrt{2}( \ln|\sqrt{8} - 2| - \ln|\sqrt{6} - 2| ) = 2\sqrt{2} \ln \left( \frac{\sqrt{8} - 2}{\sqrt{6} - 2} \right) \]This simplifies to:\[ 2\sqrt{2} \ln \left( \frac{2\sqrt{2} - 2}{\sqrt{6} - 2} \right) \]

Key concepts

Integration Techniques

Integration techniques are various strategies or methods used to find the integral of functions. These techniques help simplify and systematically solve complex integrals.
Here are a few common techniques:

• **Substitution Method:** Replaces a part of the integral with a single variable to simplify the expression.
• **Integration by Parts:** Often used when the integrand is a product of two functions.
• **Partial Fraction Decomposition:** Useful for integrals with rational functions.
• **Trigonometric Integrals:** These involve trigonometric identities to integrate expressions.

For our problem, we focus on substitution, as indicated by the presence of a square root, which can often be resolved using a simple substitution.
This will convert the integral into a more manageable form.

Definite Integrals

Definite integrals calculate the net area under the curve of a function between two specific points. These points are called bounds.
Definite integrals are expressed in the form:\[\int_{a}^{b} f(x) \, dx\]where \(a\) and \(b\) are the lower and upper bounds, respectively. The result is a number representing the area.
When solving, ensure you:

• Find the antiderivative of the integrand.
• Apply the Fundamental Theorem of Calculus by evaluating this antiderivative at the upper bound and then the lower bound.
• Subtract the two results to find the net area.

For our original integral, we converted it first and then used these principles to evaluate from \(t = 3\) to \(t = 4\).

Substitution Method

The substitution method is integral to solving many calculus problems as it simplifies complex expressions by changing variables. Here’s how it works:

• Select a substitution variable that simplifies the expression.
• Replace all instances of the original variable in the integral.
• Adjust the differential accordingly.
• Change the limits of integration to reflect the substitution.

In the given exercise, we used the substitution \(u = \sqrt{2t}\). This transformed \(dt\) and the integrand. Adjusting the limits of integration was also necessary to accommodate this change, providing a new integral that is simpler to evaluate.
Substitution is a powerful tool, commonly used when you spot derivatives within the integrand or a form resembling known integral results.

Logarithmic Integration

Logarithmic integration is used when dealing with integrals of the form \( \int \frac{1}{x} \, dx \). The result involves natural logarithms, expressed as \( \ln|x| + C \) where \( C \) is a constant.
This technique applies beyond direct forms. For example, any rational function simplified to form \(\frac{1}{u}\) suggests a logarithmic integral. That's what made it possible in our exercise to integrate \( \frac{2\sqrt{2}}{u - 2} \).
The general steps include:

• Identify if the integrand can be simplified to resemble \( \frac{1}{x} \).
• Rewrite the integral, ensuring it fits a known logarithmic form.
• Integrate using the natural log, applying bounds if definite.

In the example, the expression was simplified so that the logarithmic identity could be applied, yielding the correct antiderivative.