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.

∫ (x² dx) / (4 + x²)

Accepted Answer

Welcome back everyone. Find the integral of X2 divided by X2 + 16 DX. So for this problem, let's perform division, and we can actually perform it in a simpler way. We can add and subtract 16 and the numerator, which gives us x2 + 16 minus 16 divided by x2 + 16, and this is basically equal to 1 minus 16 divided by x2 + 16. So we got our remainder and we can basically integrate 1 minus 16 divided by x2 + 16. D X We can now write two integrals. One of them is 1 dx or DX. -16 integral of dx divided by x2 + 16. So now for the first integral we simply get the integral of dx which is x, and we're going to subtract 16 multiplied by the integral of dx divided by x2 + 16, and this is also an elementary integral. Let's remember that the integral of dx divided by x2 plus a 2 is equal to 1 divided by a. Inverse of tangent of x divided by a plus a constant of integration c. So in this case, A is equal to 4 because A2 is equal to 16 and we're taking the positive A value, right? So a squared is 16, a is 4, and we basically get the following 1 divided by a, which is 1 divided by 4 inverse of tangent of x divided by 4 plus c. So now 16 divided by 4 is 4. We can simplify slightly. And that's how we get our final answer, x minus 4, inverse of tangent of x divided by 4 plus C. Let's label it and 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.∫ (x² dx) / (4 + x²)

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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.
∫ (x² dx) / (4 + x²)