Questions: ∫ from 0 to π / 2 e^(sin(x)) cos(x) dx =

Solution Steps

To evaluate the definite integral \(\int_{0}^{\pi / 2} e^{\sin (x)} \cos (x) \, dx\), we can use a substitution method. Let \(u = \sin(x)\). Then, \(du = \cos(x) \, dx\). The limits of integration will change accordingly: when \(x = 0\), \(u = \sin(0) = 0\); and when \(x = \pi/2\), \(u = \sin(\pi/2) = 1\). The integral then simplifies to \(\int_{0}^{1} e^u \, du\), which is straightforward to evaluate.

To evaluate the definite integral \(\int_{0}^{\pi / 2} e^{\sin (x)} \cos (x) \, dx\), we use the substitution \(u = \sin(x)\). Then, \(du = \cos(x) \, dx\).

The limits of integration change accordingly:

• When \(x = 0\), \(u = \sin(0) = 0\).
• When \(x = \pi/2\), \(u = \sin(\pi/2) = 1\).

The integral simplifies to: \[ \int_{0}^{1} e^u \, du \]

The integral of \(e^u\) with respect to \(u\) is \(e^u\). Evaluating this from 0 to 1, we get: \[ \left[ e^u \right]_{0}^{1} = e^1 - e^0 = e - 1 \]

The numerical value of \(e - 1\) is approximately 1.7183.

Final Answer