What is the integral of rivative of sin(x)+10 ?

The integral of rivative of sin(x)+10 is x derivative of sin(x)+10x-xcos(x)+sin(x)+C

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integral of rivative of sin(x)+10

\int \frac{d}{d x} (sin (x)) + 10 d x

x \frac{d}{d x} (sin (x)) + 10 x - x cos (x) + sin (x) + C

Solution steps

\int \frac{d}{d x} (sin (x)) + 10 d x

Apply Integration By Parts

: x (\frac{d}{d x} (sin (x)) + 10) - \int - x sin (x) d x

= x (\frac{d}{d x} (sin (x)) + 10) - \int - x sin (x) d x

\int - x sin (x) d x = x cos (x) - sin (x)

= x (\frac{d}{d x} (sin (x)) + 10) - (x cos (x) - sin (x))

Simplify

x (\frac{d}{d x} (sin (x)) + 10) - (x cos (x) - sin (x)) : x \frac{d}{d x} (sin (x)) + 10 x - x cos (x) + sin (x)

= x \frac{d}{d x} (sin (x)) + 10 x - x cos (x) + sin (x)

Add a constant to the solution

= x \frac{d}{d x} (sin (x)) + 10 x - x cos (x) + sin (x) + C

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What is the integral of rivative of sin(x)+10 ?
The integral of rivative of sin(x)+10 is x derivative of sin(x)+10x-xcos(x)+sin(x)+C