Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.57. ∫ (from 0 to √3/2) 4/(9 + 4x²) dx

Accepted Answer

Step 1: Recognize the integral's form. The integrand resembles the formula for the derivative of the arctangent function: $$ \frac{1}{a^2 + x^2} . $$Specifically, $$ \int \frac{1}{a^2 + x^2} dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C . $$Here, the denominator is $$ 9 + 4x^2 $$, which can be rewritten as $$ (3)^2 + (2x)^2 . $$Step 2: Factor out constants to simplify the integral. Rewrite the integrand as $$ \frac{4}{9 + 4x^2} = \frac{4}{(3)^2 + (2x)^2} . $$Factor out $$ 4 $$ from the numerator and denominator: $$ \int \frac{4}{(3)^2 + (2x)^2} dx = 4 \int \frac{1}{(3)^2 + (2x)^2} dx . $$Step 3: Perform a substitution to simplify further. Let $$ u = 2x $$, so $$ du = 2 dx . $$This substitution transforms the integral into $$ 4 \int \frac{1}{(3)^2 + u^2} \cdot \frac{du}{2} $$, which simplifies to $$ 2 \int \frac{1}{(3)^2 + u^2} du . $$Step 4: Apply the arctangent formula. Using $$ \int \frac{1}{a^2 + u^2} du = \frac{1}{a} \arctan\left(\frac{u}{a}\right) + C $$, where $$ a = 3 $$, the integral becomes $$ 2 \cdot \frac{1}{3} \arctan\left(\frac{u}{3}\right) + C $$, or $$ \frac{2}{3} \arctan\left(\frac{u}{3}\right) + C . $$Step 5: Substitute back $$ u = 2x $$ and evaluate the definite integral. Replace $$ u $$ with $$ 2x  to $$get $$ \frac{2}{3} \arctan\left(\frac{2x}{3}\right) . $$Evaluate this expression at the bounds $$ x = 0 $$ and $$ x = \frac{\sqrt{3}}{2}  to $$find the final result.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
57. ∫ (from 0 to √3/2) 4/(9 + 4x²) dx

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