Question

Find the indefinite integral and check the result by differentiation. $$ \int\left(2 \sin x-5 e^{x}\right) d x $$

Short answer

The indefinite integral of the function \(2 \sin x-5 e^{x}\) is \(-2\cos x - 5e^{x} + C\), where C is the constant of integration.

Step-by-step solution

Split the Integral

The integral can be split as the sum of two separate integrals. Thus, we have \( \int(2 \sin x-5 e^{x}) dx = \int2 \sin x dx - \int5e^{x}dx \)

Apply the Basic Integration Rules

The integral of \(\sin x\) is \(-\cos x) and for \(e^{x}\) it is \(e^{x}\). Remember to apply the constants in front of these functions. Hence, the solution will be: \( -2\cos x - 5e^{x} + C\), where \( C \) is the constant of integration.

Check Result by Differentiating

Now, differentiate the obtained result to check the solution. The derivative of \(-2\cos x\) is \(2\sin x\) and the derivative of \(-5e^{x}\) is \(-5e^{x}\). Thus, differentiating the result gets us back to the original integrand \(2 \sin x-5 e^{x}\), hence confirming that the solution is correct.