Calculus 5.5 U Substitution - Definite Integrals.

Question

I need help solving a u sub definite integral question. I'll give a sample similar to it.

∫ b value is pi/2 and a value is -pi/2 the function is (4x^3 * sin(4x))/(1+x^6) dx

Answer 1

Let f(x) = (5x⁴sin(5x)) / (1 + x⁸). Recall that sin(-θ) = -sinθ.

f(-x) = -f(x), so f is an odd function. Therefore, the graph of y = f(x) is symmetric with respect to the origin.

For any odd function, ∫_{(from -a to a)} f(x)dx = 0. This is because the areas below the x-axis and above the x-axis offset.

So, ∫ _{(from -π/2 to π/2)} [(5x⁴sin(5x) / (1+x⁸)]dx = 0

You don't need u-substitution at all for this problem!