Question

А)Integral e^5x + 1 dx; б)Integral 3dx/cos^2(3x-1)

Answer 1

Во вложении/////////////////////

Answer 2

a) To integrate e^(5x+1) dx, we can use u-substitution. Let u = 5x+1, then du/dx = 5 and dx = du/5. Substituting these into the integral, we get:

∫e^(5x+1) dx = (1/5) ∫e^u du

Integrating e^u with respect to u, we get:

∫e^u du = e^u + C

where C is the constant of integration. Substituting back u = 5x+1, we get:

∫e^(5x+1) dx = (1/5) e^(5x+1) + C

Therefore, the antiderivative of e^(5x+1) is (1/5) e^(5x+1) + C.

b) To integrate 3dx/cos^2(3x-1), we can use the trigonometric identity sec^2(x) = 1/cos^2(x). Therefore, we can rewrite the integral as:

∫3 sec^2(3x-1) dx

Using the u-substitution method, let u = 3x-1, then du/dx = 3 and dx = du/3. Substituting these into the integral, we get:

∫3 sec^2(3x-1) dx = ∫sec^2(u) (du/3) = (1/3) tan(u) + C

where C is the constant of integration. Substituting back u = 3x-1, we get:

∫3dx/cos^2(3x-1) = (1/3) tan(3x-1) + C

Therefore, the antiderivative of 3dx/cos^2(3x-1) is (1/3) tan(3x-1) + C.