Question

Give an expression for p(x) so the integral can be evaluated useing substitution do not evaluate.

intergal p(x)e^x^4 dx


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Answer 1

To evaluate the integral of p(x)e^x^4 dx using substitution, p(x) should be chosen such that it allows for the substitution u = x^4. p(x) could be 4x^3 or a constant multiple of x^3 to facilitate this process.

Step-by-step explanation:

To solve the integral of p(x)ex^4 dx using substitution, we need to choose a substitution that simplifies the integral. A suitable substitution here is u = x4, with du = 4x^3.dx. The function p(x) should simplify this relationship such that when we replace x^3 with a term from p(x), the resulting expression allows for an easy substitution, turning the integral into one involving just e^u and du.

Therefore, an appropriate expression for p(x) would be one that includes the term 4x^3 or something directly proportional to it. For example, if we let p(x) = 4x^3 or p(x) = Cx^3 where C is a constant, then after factorizing we would obtain a new integral of the form Ceudu which would be straightforward to integrate using substitution.

Answer 2

4x^3

Step-by-step explanation:

Choose p so that it's the derivative of x^4.

This would give us integral(e^u du) which is integratable over elementary functions....simply it is e^u+c.

Anyways, p=(x^4)' which gives 4x^3 by power rule.

Choose p(x)=4x^3 or a constant multiple of this would also suffice.