Find all the antiderivatives of the following function. Check your work by taking derivatives. g(x)=−4cos⁡x−x

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Find all the antiderivatives of the following function. Check your work by taking derivatives.  g(x)=−4cos⁡x−x

likvau · Accepted Answer

Step 1  To find antiderivative of function g(x)=−4cos⁡x−x  Solution:  Antiderivative means integration.  We have to integrate the given function.  Integrating the given function.  ∫[−4cos⁡x−x]dx=∫−4cos⁡xdx−∫xdx  =−4∫cos⁡xdx−∫xdx  =−4sin⁡x−x22+C  Therefore, antiderivative of the function is [−4sin⁡x−x22+C]  Step 2  Verification:  If we differentiate the antiderivative of the function then we should get back original function.  Differentiating the antiderivative of the function.  ddx[−4sin⁡x−x22+C]=ddx(−4sin⁡x)−ddx(x22)+ddx(C)  =−4cos⁡x−2x2+0  =−4cos⁡x−x  We get the function back on differentiating.  Therefore, our answer is correct.  Hence, verified.

Find all the antiderivatives of the following function. Check your work by taking derivatives. g(x)=-4\cos x-x

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