Antiderivative of: t ↦ ( 1 − t 2 ) λ I would like to find an antiderivative of the function t ↦ ( 1 − t 2 ) λ where λ ∈ R &gt; 0 .I really don't know how to proceed. One idea is to use the generalized binomial theorem to get: ( 1 − t 2 ) λ = ∑ k = 0 ∞ ( λ k ) ( − 1 ) k t 2 k And by termwise integration I get that a possible antiderivative is ∑ k = 0 ∞ ( λ k ) ( − 1 ) k t 2 k + 1 2 k + 1 .The problem is that this form isn't really helpful. So is there a close form of this? So that I can study the behavior of the function when λ → ∞ on [0,1], for example.

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Antiderivative of:   t  ↦  (  1  −      t    2        )    λ  I would like to find an antiderivative of the function   t  ↦  (  1  −      t    2        )    λ   where   λ  ∈            R              &amp;gt;      0      .I really don&#039;t know how to proceed. One idea is to use the generalized binomial theorem to get:  (  1  −      t    2        )    λ    =      ∑          k      =      0              ∞                          (                    λ      k                      )              (  −  1      )    k        t          2      k      And by termwise integration I get that a possible antiderivative is       ∑          k      =      0              ∞                          (                    λ      k                      )              (  −  1      )    k              t              2        k        +        1                    2      k      +      1      .The problem is that this form isn&#039;t really helpful. So is there a close form of this? So that I can study the behavior of the function when   λ  →  ∞ on [0,1], for example.

imperiablogyy · Accepted Answer

Step 1It turns out that no elementary function is the antiderivative you are looking for. The simplest answer makes use of the hypergeometric function:  f  (  t  )  =              2        F    1        (          1      2        ,    −          λ        ,          3      2        ,    −          t      2        )    ⋅  t  +  CStep 2The differential equation satisfied by               2        F    1   will then give       f    ′    (  t  )  =  (  1  −      t    2        )    λ  .You can then get a series around   t  =  0 to study the effect of   λ. Intuitively, you&#039;d expect (t) to go to infinity as   λ  →  ∞ for every non-zero t.

Kamden Larson · Accepted Answer

Explanation:Let   t  =  sin  ⁡  θ and substitute to get  ∫  (  cos  ⁡  θ      )          2      λ      +      1          d  θ  ..This is still not a pretty anti-derivative (especially if λ is not an integer) but you know exactly what the graph looks like , so you can study the behavior as   λ  →  ∞, perhaps.

Antiderivative of: t rightarrow (1-t^{2})^{lambda)

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New Questions in Calculus 1