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SeReNiTy
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can anyone help me find an antiderivative for (x^2)((9-(x^2))^(1/2))
HallsofIvy said:When you see [itex]\sqrt{1- x^2}[/itex] or anything like that, you should think [itex]cos(x)=\sqrt{1- sin^2(x)}[/itex]- and use a trig substitution.
In this problem, factor a "9" out of the squareroot to get [itex]3x^2\sqrt{1- \frac{x^2}{9}}[/itex]. Now make the substitution x= 3sin(θ).
dx= 3cos(θ)dθ and [itex]\sqrt{1- \frac{x^2}{9}}[/itex] becomes [itex]\sqrt{1- sin^2(\theta)}= cos(\theta)[/itex]. The entire integrand becomes sin2(θ)cos2(θ)dθ. You will need to use trig substitutions to integrate that.
The antiderivative is a fundamental concept in calculus that represents the inverse operation of differentiation. It is essentially the process of finding a function that, when differentiated, gives the original function.
To solve this expression, you first need to rewrite it as a product of two functions, one of which can be integrated easily. In this case, you can rewrite it as x^2 * (9-x^2)^1/2. Then, you can use the power rule for integration to solve it.
The power rule for integration states that the integral of x^n is equal to (1/(n+1)) * x^(n+1), where n is any real number except for -1. This rule is useful for finding the antiderivative of polynomials.
No, the power rule for integration only applies to functions that can be written as a polynomial or a product of a polynomial and a constant. For other functions, you may need to use other integration techniques such as substitution or integration by parts.
Finding antiderivatives is important in calculus as it allows us to solve problems involving the accumulation of quantities over time or distance. It is also a crucial step in finding definite integrals, which are used to solve a variety of real-world problems in fields such as physics, economics, and engineering.