What is the integral of cos^2x-e^x with respect to x ?

Question

What is the integral  of cos^2x-e^x with respect to x ?

Bytelearn · Accepted Answer

Break down the integral: Break down the integral into two separate integrals. We have the integral of cos^2(x) - e^x, which can be written as the integral of cos^2(x) minus the integral of e^x. ∫(cos^2(x) - e^x) dx = ∫cos^2(x) dx - ∫e^x dxIntegrate cos^2(x): Integrate the first part, ∫cos^2(x) dx. To integrate cos^2(x), we use the power-reduction formula, which states that cos^2(x) = (1 + cos(2x))/2. So, ∫cos^2(x) dx = ∫(1/2 + 1/2 cos(2x)) dxIntegrate terms separately: Integrate the terms separately. Now we integrate 1/2 and 1/2 cos(2x) separately. ∫(1/2 + 1/2 cos(2x)) dx = 1/2 ∫dx + 1/2 ∫cos(2x) dx = 1/2 x + 1/4 sin(2x) + C_1, where C_1 is the constant of integration for this part.Integrate e^x: Integrate the second part, ∫e^x dx. The integral of e^x with respect to x is simply e^x. So, ∫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. Now we combine the integrals of both parts to get the final answer. ∫(cos^2(x) - e^x) dx = (1/2 x + 1/4 sin(2x) + C_1) - (e^x + C_2)Simplify expression: Simplify the expression and combine the constants of integration. We can combine C_1 and C_2 into a single constant of integration, C. So, the final answer is: ∫(cos^2(x) - e^x) dx = 1/2 x + 1/4 sin(2x) - e^x + C

What is the integral of $\cos^2 x - e^x$ with respect to $x$ ?

Full solution

More problems from Solve two-step inequalities

Related topics

What is the integral of cos⁡2x−ex\cos^2 x - e^xcos2x−ex with respect to xxx?

What is the integral of $\cos^2 x - e^x$ with respect to $x$ ?