Which of the following integrals cannot be evaluated using a simple substitution?
the integral of 1 divided by the quantity x squared plus 1, dx
the integral of 1 divided by the quantity x squared plus 1, dx
the integral of x divided by the quantity x squared plus 1, dx
(MY ANSWER)
the integral of x cubed divided by the quantity x to the 4th power plus 1, dx
To determine which of the integrals cannot be evaluated using a simple substitution, we need to analyze the complexity and pattern of the integrands.
For the first integral, the integrand is 1/(x^2 + 1). This can be evaluated through a simple substitution, letting u = x^2 + 1, which transforms the integral into ∫(1/u) du.
For the second integral, the integrand is also 1/(x^2 + 1), which is the same as the first integral. Therefore, this integral can also be evaluated using a simple substitution.
For the third integral, the integrand is x/(x^2 + 1). You can try a simple substitution of u = x^2 + 1, but it will not simplify the integral. Therefore, this integral cannot be evaluated using a simple substitution.
For the fourth integral, the integrand is x^3/(x^4 + 1). Although it appears complex, it can be simplified using a simple substitution. Let u = x^4 + 1, which transforms the integral into ∫(1/u) du. Therefore, this integral can also be evaluated using a simple substitution.
Based on our analysis, the third integral, the integral of x/(x^2 + 1), cannot be evaluated using a simple substitution.
Nope.
∫ 1/(x^2+1) dx :: not so simple here. You need x = tanθ
The 2nd is the same, unless there are some groupings you have omitted.
∫ x/(x^2+1) dx :: u = x^2+1
∫ x^3/(x^4+1) dx :: u = x^4+1
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