Find the indefinite integral and check the result by differentiation. (Use C for the constant of...

Question:

Find the indefinite integral and check the result by differentiation. (Use C for the constant of integration.)

{eq}\displaystyle \int (8 \sin x - 3 e^x)\ dx {/eq}.

Insert context header here:

{eq}\int ( \sin x)\ dx=-cosx\\ \int( e^x)\ dx=e^x\\ \frac{d(cosx)}{dx}=-sinx\\ \frac{d(e^x)}{dx}=e^x\\ {/eq}

Answer and Explanation:

{eq}\text{We have to integrate }\int (8 \sin x - 3 e^x)\ dx {/eq}

$$\int (8 \sin x - 3 e^x)\ dx\\ \text{Apply linearity}\\ 8\int ( \sin x)\ dx - 3\int( e^x)\ dx\\ \text{Now solving}\\ 8\int ( \sin x)\ dx\\ \text{This is standard integral}\\ -8cosx\\ \text{Now solving}\\ 3\int( e^x)\ dx\\ \text{This is standard integral}\\ 3e^x\\ \text{Hence, }\int (8 \sin x - 3 e^x)\ dx\\=-8cosx-3e^x+c $$

{eq}\text{For checking our integration we have to differentiate }-8cosx-3e^x+c {/eq}

$$\frac{d(-8cosx-3e^x+c)}{dx}\\ \text{we know that }\frac{d(cosx)}{dx}=-sinx\;\;and\;\;\frac{d(e^x)}{dx}=e^x\\ \Rightarrow-8(-sinx)-3e^x\\ \Rightarrow8sinx-3e^x\\ \text{Here we get the expression we integrated. Hence, the result is checked} $$


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