Which of the following functions of x can be integrated with respect to x without integration by parts, and purely using integration by u-substitution and the knowledge of the antiderivative of the tangent function?

Question

Which of the following functions of x can be integrated with respect to x without integration by parts, and purely using integration by u-substitution and the knowledge of the antiderivative of the tangent function?

a) x^2tan(x^2)
b) xtan^2(x)
c) x^2tan(x)
d) xtan(x^2)

Answer 1

looks like D to me

u = x^2

Answer 2

let u = x^2

du = 2x dx so dx = du/2x
try d)
int x tan x^2 dx = int x tan u du/(2x)
= int (1/2) tan u du

Answer 3

Sorry, was drinking coffee while oobleck did it :)

Answer 4

To determine which of the given functions can be integrated using u-substitution and the antiderivative of the tangent function, let's analyze each option.

a) x^2tan(x^2)
To integrate this function, u-substitution can be employed. Let u = x^2, and then du = 2xdx. The function can be rewritten as:
∫ x^2tan(x^2) dx = (1/2) ∫ tan(u) du

The antiderivative of tan(u) is -ln|cos(u)|. Therefore:
(1/2) ∫ tan(u) du = (1/2)(-ln|cos(u)|) + C
= -(1/2)ln|cos(x^2)| + C

b) xtan^2(x)
To integrate this function, u-substitution is not directly applicable. However, since tan^2(x) = sec^2(x) - 1, we can rewrite the function as:
xtan^2(x) = x(sec^2(x) - 1)

Now, we can integrate by using u-substitution with u = sec(x), and then du = sec(x)tan(x)dx. The function can be expressed as:
∫ x(sec^2(x) - 1) dx = ∫ (u^2 - 1) du

The antiderivative of (u^2 - 1) is (1/3)u^3 - u. Substituting u back to sec(x):
∫ (u^2 - 1) du = (1/3)sec^3(x) - sec(x) + C

c) x^2tan(x)
Similar to the first option, this function can be integrated using u-substitution. Let u = x, and then du = dx. The function becomes:
∫ x^2tan(x) dx = ∫ u^2tan(u) du

As we know that the antiderivative of tan(u) is -ln|cos(u)|:
∫ u^2tan(u) du = -(1/2)u^2ln|cos(u)| + (1/2) ∫ u^2 du
= -(1/2)u^2ln|cos(u)| + (1/2)(u^3/3) + C

d) xtan(x^2)
For this function, u-substitution does not directly apply. Integration by parts is required, which involves taking part of the function as u and the remaining as dv. However, this option does not fall strictly under the criteria of using only u-substitution and the antiderivative of the tangent function, so it cannot be integrated using those methods alone.

In conclusion, the functions that can be integrated purely with u-substitution and the antiderivative of the tangent function are a) x^2tan(x^2) and c) x^2tan(x).