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What is the integral of with respect to ?
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Q. What is the integral of with respect to
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- Break down the integral: Break down the integral into two separate integrals.We have the integral of , which can be written as the integral of minus the integral of .$\int(\cos^\(2\)(x) - e^x) \, dx = \int\cos^\(2\)(x) \, dx - \int e^x \, dx
- Integrate \(\cos^2(x)\): Integrate the first part, \(\int \cos^2(x) \, dx\). To integrate \(\cos^2(x)\), we use the power-reduction formula, which states that \(\cos^2(x) = \frac{1 + \cos(2x)}{2}\). So, \(\int \cos^2(x) \, dx = \int\left(\frac{1}{2} + \frac{1}{2} \cos(2x)\right) dx\)
- Integrate terms separately: Integrate the terms separately.\(\newline\)Now we integrate \(\frac{1}{2}\) and \(\frac{1}{2} \cos(2x)\) separately.\(\newline\)\(\int(\frac{1}{2} + \frac{1}{2} \cos(2x)) \, dx = \frac{1}{2} \int dx + \frac{1}{2} \int \cos(2x) \, dx\)\(\newline\)\(= \frac{1}{2} x + \frac{1}{4} \sin(2x) + C_1\), where \(C_1\) is the constant of integration for this part.
- Integrate \(e^x\): Integrate the second part, \(\int e^x \, dx\). The integral of \(e^x\) with respect to \(x\) is simply \(e^x\). So, \(\int e^x \, dx = e^x + C_2\), where \(C_2\) is the constant of integration for this part.
- Combine results: Combine the results from Step \(3\) and Step \(4\).\(\newline\)Now we combine the integrals of both parts to get the final answer.\(\newline\)\(\int(\cos^2(x) - e^x) \, dx = (\frac{1}{2} x + \frac{1}{4} \sin(2x) + C_1) - (e^x + C_2)\)
- Simplify expression: Simplify the expression and combine the constants of integration.\(\newline\)We can combine \(C_1\) and \(C_2\) into a single constant of integration, \(C\).\(\newline\)So, the final answer is:\(\newline\)\(\int(\cos^2(x) - e^x) dx = \frac{1}{2} x + \frac{1}{4} \sin(2x) - e^x + C\)
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