Mha sleepover

*you see sero and denki on the bed, and sero was holding denki down and covering his mouth. Denki was trying to get free*

*you and class 1-A was having a sleepover on the main room* *mina, momo, uraraka, tsyu, and jiro was making cookies while the boys were playing video games* denki: damn sero…..your baaaad at this.. *makes a sheep joke* sero: alright, I’m done with this. *he grabs denki and takes him to his room* deku: I hope he’s okay…. Mineta: relax….we can just watch all the girls and- *mina throws a cookie at him*

Umuru: bruh im bo- what what *the is clapping sound from sero room*

Denki: W-WHAT THE HELL IS GOING ON?!? *gets out of sero's room and sees the girls all giggling*

To evaluate the integral \(\int_{0}^{\infty} e^{-x} \cos(x) \, dx\), we can use integration by parts twice and solve for the integral. Here's a step-by-step solution: 1. **First Integration by Parts:** - Let \( u = \cos(x) \), \( dv = e^{-x} dx \) - Then \( du = -\sin(x) dx \), \( v = -e^{-x} \) \[ \int e^{-x} \cos(x) dx = -e^{-x} \cos(x) - \int e^{-x} \sin(x) dx \] 2. **Second Integration by Parts:** - For \(\int e^{-x} \sin(x) dx\), let \( u = \sin(x) \), \( dv = e^{-x} dx \) - Then \( du = \cos(x) dx \), \( v = -e^{-x} \) \[ \int e^{-x} \sin(x) dx = -e^{-x} \sin(x) + \int e^{-x} \cos(x) dx \] 3. **Combine Results:** - Substitute back into the first equation: \[ \int e^{-x} \cos(x) dx = -e^{-x} \cos(x) - \left[ -e^{-x} \sin(x) + \int e^{-x} \cos(x) dx \right] \] - Let \( I = \int e^{-x} \cos(x) dx \), then: \[ I = -e^{-x} \cos(x) + e^{-x} \sin(x) - I \] \[ 2I = e^{-x} (\sin(x) - \cos(x)) \] \[ I = \frac{e^{-x} (\sin(x) - \cos(x))}{2} + C \] 4. **Evaluate Definite Integral:** - Compute from \( 0 \) to \( \infty \): \[ \left. \frac{e^{-x} (\sin(x) - \cos(x))}{2} \right|_{0}^{\infty} \] - As \( x \to \infty \), \( e^{-x} \to 0 \), so the upper limit term is \( 0 \). - At \( x = 0 \): \[ \frac{e^{0} (0 - 1)}{2} = -\frac{1}{2} \] - Subtract lower limit from upper limit: \[ 0 - \left(-\frac{1}{2}\right) = \frac{1}{2} \] **Answer:** \(\boxed{\dfrac{1}{2}}\)

*you and class 1-A was having a sleepover on the main room* *mina, momo, uraraka, tsyu, and jiro was making cookies while the boys were playing video games* denki: damn sero…..your baaaad at this.. *makes a sheep joke* sero: alright, I’m done with this. *he grabs denki and takes him to his room* deku: I hope he’s okay…. Mineta: relax….we can just watch all the girls and- *mina throws a cookie at him*

damn denki is gonna be filled with white cream