Use any method to evaluate the integrals in Exercises 15–38. M...

Question

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (x dx) / (25 + 4x²)

Accepted Answer

Welcome back, everyone. Evaluate the integral XDX divided by 9 + 2 X squared. For this problem, let's go ahead and use U substitution. Let's suppose that U is equal to our denominator, 9 + 2 X2. What we have to do from here is identify the derivative DU divided by D X. The derivative is going to be 0 + 4 X, and now we can multiply both sides by DX and divide by 4 to show that 1/4 DU is equal to XDX. This is exactly what we have in the numerator, meaning we can now rewrite our integral as integral of 1/4 DU because that's XDX, and in the denominator we have U. We can factor out 14th and integrate the EU divided by U which is an elementary integral. Its value is LN of the absolute value of U plus C. We're multiplying that by 1/4. And now, we can remove the absolute value because U is greater than 0 for any X. We have a positive number plus a non-negative number because X2d has a minimum value of 0. So now we're going to replace you with the actual expression in terms of X to get 1/4 LN of 9 + 2 X2 + C, which is our final answer for this problem. Thank you for watching.

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.∫ (x dx) / (25 + 4x²)

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Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x dx) / (25 + 4x²)