Chapter 17: Problem 2

Find $\int(x+3) \sin x \mathrm{~d} x$.

Short Answer

Expert verified

Question: Find the integral of the function $(x+3)\sin x$. Answer: The integral of $(x+3)\sin x$ is $-(x+3)\cos x + \sin x + C$, where C is the constant of integration.

Step by step solution

Identify u and dv

First, we need to identify two functions $u$ and $dv$ within the integral. The integral is given as $\int (x+3) \sin x \mathrm{~d} x$. We can select $u$ as $(x+3)$ and $dv$ as $\sin x \mathrm{~d} x$. This choice is made because it is easier to differentiate $(x+3)$ and integrate $\sin x$.

Find du and v

To apply integration by parts, we need to find $du$ and $v$ from $u$ and $dv$. Differentiating our chosen $u=(x+3)$ with respect to $x$ gives $du=(1) \mathrm{~d} x$. Integrating our chosen $dv=\sin x \mathrm{~d} x$ gives $v=-\cos x$.

Apply Integration by Parts

Now we apply the integration by parts rule $\int u \; dv = uv - \int v \; du$. We have: $$\int(x+3) \sin x \mathrm{~d} x = (x+3)(-\cos x) - \int (-\cos x)(1) \mathrm{~d} x$$

Integrate v du

Now we need to integrate the remaining integral on the right side: $$\int (-\cos x)(1) \mathrm{~d} x = -\int \cos x \mathrm{~d} x$$ Integrating $\cos x$ with respect to $x$ gives $\sin x$. Thus the integral becomes: $$-\int \cos x \mathrm{~d} x = -\sin x$$

Combine Results

Substitute the result of integrating $-\cos x$ back into the expression obtained in Step 3: $$(x+3)(-\cos x) - \int (-\cos x)(1) \mathrm{~d} x = (x+3)(-\cos x) - (-\sin x)$$

Simplify Expression

Simplify the expression to obtain the final answer: $$\int (x+3) \sin x \mathrm{~d} x = (x+3)(-\cos x) + \sin x + C$$ The final result is: $$\int (x+3) \sin x \mathrm{~d} x = -(x+3)\cos x + \sin x + C$$

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Find \(\int(x+3) \sin x \mathrm{~d} x\).

Short Answer

Step by step solution

Identify u and dv

Find du and v

Apply Integration by Parts

Integrate v du

Combine Results

Simplify Expression

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