To solve the definite integral [tex]\(\int_0^{\frac{\pi}{4}} (\sin(y) + \sec^2(y)) \, dy\)[/tex], we will follow these steps.
1. Break Down the Integral:
The given integral can be split into two separate integrals:
[tex]\[
\int_0^{\frac{\pi}{4}} (\sin(y) + \sec^2(y)) \, dy = \int_0^{\frac{\pi}{4}} \sin(y) \, dy + \int_0^{\frac{\pi}{4}} \sec^2(y) \, dy
\][/tex]
2. Evaluate Each Integral Separately:
a. The first integral is:
[tex]\[
\int_0^{\frac{\pi}{4}} \sin(y) \, dy
\][/tex]
The antiderivative of [tex]\(\sin(y)\)[/tex] is [tex]\(-\cos(y)\)[/tex]. Therefore:
[tex]\[
\int_0^{\frac{\pi}{4}} \sin(y) \, dy = \left[-\cos(y)\right]_0^{\frac{\pi}{4}}
\][/tex]
Evaluating this at the bounds, we get:
[tex]\[
-\cos\left(\frac{\pi}{4}\right) + \cos(0)
\][/tex]
Since [tex]\(\cos\left(\frac{\pi}{4}\right) = \frac{1}{\sqrt{2}}\)[/tex] and [tex]\(\cos(0) = 1\)[/tex], we have:
[tex]\[
-\frac{1}{\sqrt{2}} + 1 = 1 - \frac{1}{\sqrt{2}}
\][/tex]
b. The second integral is:
[tex]\[
\int_0^{\frac{\pi}{4}} \sec^2(y) \, dy
\][/tex]
The antiderivative of [tex]\(\sec^2(y)\)[/tex] is [tex]\(\tan(y)\)[/tex]. Therefore:
[tex]\[
\int_0^{\frac{\pi}{4}} \sec^2(y) \, dy = \left[\tan(y)\right]_0^{\frac{\pi}{4}}
\][/tex]
Evaluating this at the bounds, we get:
[tex]\[
\tan\left(\frac{\pi}{4}\right) - \tan(0)
\][/tex]
Since [tex]\(\tan\left(\frac{\pi}{4}\right) = 1\)[/tex] and [tex]\(\tan(0) = 0\)[/tex], we have:
[tex]\[
1 - 0 = 1
\][/tex]
3. Combine the Results:
Now, we add the results of the two integrals together:
[tex]\[
\left( 1 - \frac{1}{\sqrt{2}} \right) + 1 = 2 - \frac{1}{\sqrt{2}}
\][/tex]
Simplifying further, we can express [tex]\(2 - \frac{1}{\sqrt{2}}\)[/tex] in a more accurate numerical form. Using decimal approximations:
[tex]\[
\frac{1}{\sqrt{2}} \approx 0.7071067811865475
\][/tex]
So,
[tex]\[
2 - 0.7071067811865475 \approx 1.29289321881345
\][/tex]
Thus, the value of the definite integral [tex]\(\int_0^{\frac{\pi}{4}} (\sin(y) + \sec^2(y)) \, dy\)[/tex] is approximately [tex]\(1.29289321881345\)[/tex].
