Question

Evaluate ∫secx(2secx + 5tanx) dx, where c is the constant of integration.

A) 2tanx + 5secx + c
B) 2tanx - 5secx + c
C) -2tanx + 5secx + c
D) -2tanx - 5secx + c

Answer 1

The integral ∫secx(2secx + 5tanx) dx is evaluated by integrating each term separately, using basic trigonometric integrals to find that the result is 2tanx + 5secx + C.

Step-by-step explanation:

To evaluate the integral ∫secx(2secx + 5tanx) dx, we can look at it as integrating two separate functions, 2sec2x and 5secx tanx. The integral of sec2x is tanx, and the integral of secx tanx is secx. Both of these integrals are straightforward applications of basic trigonometric integrals.

Now, integrating term by term, we have:∫2sec2x dx = 2tanx + C1∫5secx tanx dx = 5secx + C2Adding the two results, we obtain the final answer.Integral result: 2tanx + 5secx + CTo evaluate the integral ∫secx(2secx + 5tanx) dx, we can use the substitution method. Let's substitute u = secx + tanx. Then, du = (secx * tanx + sec^2x)dx = (secx * tanx + 1)dx.The integral now becomes ∫2u du. Integrating this expression gives us the result u^2 + c, where c is the constant of integration.Substituting back u = secx + tanx, we get (secx + tanx)^2 + c as the final answer.