Please help me with this integral problem:
ʃ x^2(a^2-x^4)^2dx
please help..thanks alot:)
It's just a polynomial. Expand things out to get
ʃ x^2(a^4 - 2a^2x^4 + x^8)^2 dx
= ʃ a^4x^2 - 2a^2x^6 + x^10 dx
= a^4x^3/3 - 2a^2x^7/7 - x^11/11 + C
Expand the expression to get individual terms in x
after that it is easy
To evaluate the integral ʃ x^2(a^2-x^4)^2dx, we can expand the binomial term and simplify the expression. Here are the steps:
Step 1: Expand the binomial term.
(a^2 - x^4)^2 = (a^2 - 2*a^2*x^4 + x^8) = a^4 - 2*a^2*x^4 + x^8
Step 2: Multiply x^2 with each term of the expanded binomial.
x^2 * (a^4 - 2*a^2*x^4 + x^8) = a^4*x^2 - 2*a^2*x^6 + x^10
Now we have a polynomial expression that can be integrated term-by-term.
Step 3: Integrate term by term.
ʃ a^4*x^2 dx - ʃ 2*a^2*x^6 dx + ʃ x^10 dx
To continue integrating each term, the exponents of x become odd, allowing us to use the power rule for integration.
Step 4: Integrate each term using the power rule.
ʃ a^4*x^2 dx = (a^4/3) * x^3 + C1 (where C1 is the constant of integration)
ʃ 2*a^2*x^6 dx = (2*a^2/7) * x^7 + C2 (where C2 is the constant of integration)
ʃ x^10 dx = (1/11) * x^11 + C3 (where C3 is the constant of integration)
Step 5: Combine the separate integrals and constants of integration.
The final result of the integral is:
(a^4/3) * x^3 - (2*a^2/7) * x^7 + (1/11) * x^11 + C (where C is the constant of integration).
Therefore, the integral ʃ x^2(a^2-x^4)^2dx evaluates to:
(a^4/3) * x^3 - (2*a^2/7) * x^7 + (1/11) * x^11 + C.