Find dydx if y=∫3xx3ettantdt . For an indefinite integral, use the notation INT in place of the symbol. For example, ∫(x+1)dx should be written as INT(x+1)dx . For a definite integral, use the notation INT(a,b) . For example, ∫21(x+1)dx should be written as INT(1,2)(x+1)dx . For a definite integral that has been integrated, use the notation EVAL(a,b) . For example, (x22+x)∣∣21 should be written as (x22+x)EVAL(1,2) .
To find dy/dx, we can simply differentiate the given function y with respect to x.
y = ∫3x(x^3e^ttan(t))dt
To differentiate y with respect to x, we use the fundamental theorem of calculus:
dy/dx = d/dx [∫3x(x^3e^ttan(t))dt]
To differentiate an integral with respect to x, we use the Leibniz integral rule:
dy/dx = (d/dx [3x]) * (x^3e^xTan(x)) + [3x] * (d/dx [x^3e^xTan(x)])
Taking the derivatives:
dy/dx = (3) * (x^3e^xTan(x)) + [3x] * (3x^2e^xTan(x) + x^3e^xSec^2(x))
Simplifying:
dy/dx = 3x^3e^xTan(x) + 3x^3e^xSec^2(x) + 9x^3e^xTan(x) + 3x^4e^xSec^2(x)
Combining like terms:
dy/dx = 12x^3e^xTan(x) + 3x^4e^xSec^2(x)
Therefore, dy/dx = 12x^3e^xTan(x) + 3x^4e^xSec^2(x)