Final answer:
The integral of an odd function over a symmetric interval around zero is zero. Without further context or clarification on the integral's specifics, the answer to the integral of x⁷tan(x) from -π/4 to π/4 is 0.
Step-by-step explanation:
The student has asked to evaluate the definite integral of x³ * x⁴tan(x) from -π/4 to π/4. The integral provided in the question seems to have some typographical errors, but based on the context, it looks like we are to evaluate an integral of a function over a symmetric interval around zero.
If the integrand is an odd function (i.e., f(-x) = -f(x)), then its integral over a symmetric interval around zero is zero. In this case, if the integrand is x⁷tan(x), since both x⁷ and tan(x) are odd functions, their product is also an odd function. Therefore the integral from -π/4 to π/4 of this function is zero.
However, if there was a mistake and the integrand is meant to be different or an additional function is involved (as the provided hints might suggest), then the approach would require more information or adjustment. Similarly, if additional bounds or details were specified in the question, these would need to be taken into consideration to arrive at the correct answer.