Choose the option that gives an expression for the indefinite integral
ʃ (cos(4x) + 2x^2)(sin(4x) − x) dx.
In each option, c is an arbitrary constant.
Options
A cos(4x) + 2x^2 +c
B -1/8cos(4x) + 2x^2)^2 +c
C 1/4 (sin(4x) − x)^2 + c
D (1/(2 (sin(4x) − x)) +c
E ln cos(4x) + 2x^2 +c
F ln (sin(4x) − x) + c
G ln (cos(4x) + 2x^2)/(sin(4x) – x)+ c
H ln (2x^2 sin(4x) − x cos(4x))+ c
B in form: -1/8(cos(4x)+2x^2)^2+C
To find the expression for the indefinite integral of the given function, we can use the power rule for integration and the product rule.
The power rule states that the integral of x^n with respect to x is (1/(n+1))x^(n+1) + c, where c is an arbitrary constant.
The product rule states that the integral of the product of two functions u(x) and v(x) with respect to x is given by ʃ (u(x)v'(x) + u'(x)v(x)) dx, where u'(x) and v'(x) are the derivatives of u(x) and v(x) respectively.
Now, let's go through each option to determine which one gives the correct expression for the indefinite integral:
A. cos(4x) + 2x^2 + c
This option does not consider the other term (-x) in the integral. Therefore, it is not the correct answer.
B. (-1/8cos(4x) + 2x^2)^2 + c
This option squares the term (-1/8cos(4x) + 2x^2), which is incorrect. Also, it does not consider the other term (-x) in the integral. Therefore, it is not the correct answer.
C. 1/4 (sin(4x) − x)^2 + c
This option squares the term (sin(4x) − x), which is incorrect. Therefore, it is not the correct answer.
D. 1/(2 (sin(4x) − x)) + c
This option takes the reciprocal of the term (sin(4x) − x), which is incorrect. Therefore, it is not the correct answer.
E. ln(cos(4x) + 2x^2) + c
This option takes the natural logarithm of the term (cos(4x) + 2x^2), which is not correct because the integral of this term does not involve the natural logarithm function. Therefore, it is not the correct answer.
F. ln(sin(4x) − x) + c
This option takes the natural logarithm of the term (sin(4x) − x), which is not correct because the integral of this term does not involve the natural logarithm function. Therefore, it is not the correct answer.
G. ln((cos(4x) + 2x^2)/(sin(4x) – x)) + c
This option takes the natural logarithm of the ratio of the two terms, (cos(4x) + 2x^2) and (sin(4x) − x). This is not the correct answer as well since the integral does not involve the natural logarithm function. Therefore, it is not the correct answer.
H. ln(2x^2 sin(4x) − x cos(4x)) + c
This option takes the natural logarithm of the term (2x^2 sin(4x) − x cos(4x)). This is also incorrect because the integral does not involve the natural logarithm function. Therefore, it is not the correct answer.
Based on our analysis, the correct answer is none of the given options.