What are the limits for the integral of sin(x) * sin(x^2)?

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In summary, the integral of [sinx * sin(x^2)] can be solved using integration by parts or substitution. The indefinite integral is ∫ sinx * sin(x^2) dx = 1/2 * √(π/2) * erf(x^2 / √2) * (π/2 + 1/2 * sin(2x^2)) + C, and it can also be solved using numerical methods such as the trapezoidal rule or Simpson's rule.
  • #1
LordVader88
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I've never cracked this 1 yet, and I don't remember if it's domain was [0,1] or [0,∞), pretty sure it was 1 of the 2, and not anything with -∞ or ∏/2 for example

∫[sinx * sin(x^2)]dx
 
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  • #3
sin(x)* sin(x^2)=[sin((x-1/2)^2)-sin((x+1/2)^2)]sin(1/4)/2+[cos((x-1/2)^2)-cos((x+1/2)^2)]cos(1/4)/2

Your integral is then easily expressed in terms ofFresnel integrals. As far as what limits make the answer tidy, probably none.
 

FAQ: What are the limits for the integral of sin(x) * sin(x^2)?

What is the formula for the integral of [sinx * sin(x^2)]?

The formula for the integral of [sinx * sin(x^2)] is ∫ sinx * sin(x^2) dx = 1/2 * √(π/2) * erf(x^2 / √2) * (π/2 + 1/2 * sin(2x^2)).

What is the method for solving the integral of [sinx * sin(x^2)]?

The method for solving the integral of [sinx * sin(x^2)] is by using integration by parts. This involves breaking the integral into two parts and using the formula ∫ u * dv = u * v - ∫ v * du to solve it.

Can the integral of [sinx * sin(x^2)] be solved without using integration by parts?

Yes, the integral of [sinx * sin(x^2)] can also be solved by using the substitution method. This involves substituting x^2 with u and solving the resulting integral using the formula ∫ f(g(x)) * g'(x) dx = ∫ f(u) du.

What is the indefinite integral of [sinx * sin(x^2)]?

The indefinite integral of [sinx * sin(x^2)] is ∫ sinx * sin(x^2) dx = 1/2 * √(π/2) * erf(x^2 / √2) * (π/2 + 1/2 * sin(2x^2)) + C, where C is the constant of integration.

Can the integral of [sinx * sin(x^2)] be solved using numerical methods?

Yes, the integral of [sinx * sin(x^2)] can also be solved using numerical methods such as the trapezoidal rule or Simpson's rule. These methods involve approximating the integral by dividing it into smaller intervals and using a formula to calculate the area under the curve in each interval.

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