Question

Evaluate the integral. Evaluate the integral. (Use C for the constant of integration.) ∫ (x / (x⁴ + 16)) dx

Answer 1

The integral \(\int (x / (x^4 + 16)) dx\) is evaluated by substituting u = x^2 to simplify the integral into a standard form and then integrating using the arctangent function. The final result is \(1/8\) arctan(x^2/4) + C.

Step-by-step explanation:

To evaluate the integral \(\int (x / (x^4 + 16)) dx\), we can use a substitution method. Let's use the substitution u = x^2. Then, du = 2x dx so dx = du / (2x). This allows us to rewrite the integral in terms of u:

\(\int (x / (x^4 + 16)) dx\) becomes \(1/2 * \int (1 / (u^2 + 16)) du\). This is a standard integral that can be solved using the arctangent arctan(u/4). After we integrate, we reverse the substitution to get the solution in terms of x.

So, the integration process is:

1. Substitute u = x^2 and dx = du / (2x).
2. Compute the integral \(1/2 * \int (1 / (u^2 + 16)) du\).
3. Use the antiderivative arctan(u/4) to find the indefinite integral.
4. Reverse the substitution to replace u with x^2 in the result.
5. Add the constant of integration C.

The result is \(1/8\) arctan(x^2/4) + C.

Answer 2

The integral ∫ (x / (x⁴ + 16)) dx is (1/4) * ln |x²/4 + √(2)| + C, where C is the constant of integration. This result is obtained by utilizing a substitution, applying the inverse tangent formula, and simplifying the expression to its logarithmic form.

Step-by-step explanation:

To solve the integral ∫ (x / (x⁴ + 16)) dx, start by substituting x² = u. Therefore, dx = (1/2x) du.

The integral becomes ∫ (1 / (u² + 16)) du after substitution. Rewrite the integral as ∫ (1 / ((u²/16) + 1)) du to match the form of the inverse tangent formula: ∫ (1 / (a² + x²)) dx = (1/a) * atan(x/a) + C.

Now, we can see that u²/16 is similar to a², where a = 4. Apply the inverse tangent formula to get ∫ (1 / ((u²/16) + 1)) du = (1/4) * atan(u/4) + C.

Replace u with x², yielding (1/4) * atan(x²/4) + C. Simplify the expression using the trigonometric identity atan(x) = ln |x² + 1| / 2, obtaining (1/4) * ln |x²/4 + 1| + C.

To match the original integral, use another trigonometric identity ln(a) = 2 * ln(√a), giving (1/4) * ln |x²/4 + √(2)| + C as the final solution. Therefore, the integral of (x / (x⁴ + 16)) dx equals (1/4) * ln |x²/4 + √(2)| + C.