7–64. Integration review Evaluate the following integrals.
53. ∫ eˣ sec(eˣ + 1) dx

Guida verificata passo dopo passo

Step 1: Recognize that the integral involves a composite function, eˣ, inside another function sec(eˣ + 1). This suggests that substitution might be a useful technique.

Step 2: Let u = eˣ + 1. Then, compute the derivative of u with respect to x: du/dx = eˣ. This implies that du = eˣ dx.

Step 3: Rewrite the integral in terms of u. Substituting u = eˣ + 1 and du = eˣ dx, the integral becomes ∫ sec(u) du.

Step 4: Recall the standard integral formula for sec(u): ∫ sec(u) du = ln|sec(u) + tan(u)| + C, where C is the constant of integration.

Step 5: Substitute back u = eˣ + 1 into the result to express the solution in terms of x. The final expression will be ln|sec(eˣ + 1) + tan(eˣ + 1)| + C.

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