Question Statement

Evaluate the indefinite integral:

$\int x\cos xdx$

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Background and Explanation

This integral involves the product of an algebraic function ($x$) and a trigonometric function ($\cos x$). When integrating products of different function types, we apply integration by parts, following the LIATE priority rule (Logarithmic, Inverse, Algebraic, Trigonometric, Exponential) to select $u=x$ as the first function.

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Solution

Let $I=\int x\cos xdx$.

We apply the integration by parts formula in the form: $\int uvdx=u\int vdx-\int \left[\frac{du}{dx}\int vdx\right]dx$

Following the LIATE rule, we choose $u=x$ (algebraic) and $v=\cos x$ (trigonometric). Therefore, $\frac{du}{dx}=1$ and we proceed with the calculation:

$\begin{array}{rl}I& =x\int \cos xdx-\int \left(\int \cos xdx\right)\frac{d}{dx}(x)dx\\ & =x\sin x-\int \sin x\cdot 1dx\\ & =x\sin x-\int \sin xdx\\ & =x\sin x-(-\cos x)+c\\ & =x\sin x+\cos x+c\end{array}$

Reasoning for each step:

1. First term: Multiply $x$ by the integral of $\cos x$ (which is $\sin x$) to get $x\sin x$.
2. Second term: Subtract the integral of [the derivative of $x$ (which is $1$) multiplied by the integral of $\cos x$ (which is $\sin x$)], giving $\int \sin xdx$.
3. Evaluate the remaining integral: The integral of $\sin x$ is $-\cos x$.
4. Simplify the signs: Subtracting a negative becomes addition: $x\sin x+\cos x$.
5. Add the constant: Include $+c$ for the general indefinite integral solution.

Thus, the final answer is $x\sin x+\cos x+c$.

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Key Formulas or Methods Used

• Integration by parts: $\int uvdx=u\int vdx-\int \left[\frac{du}{dx}\int vdx\right]dx$ (equivalent to $\int udv=uv-\int vdu$)
• LIATE rule: Priority order for selecting the first function $u$ (Logarithmic $\to$ Inverse $\to$ Algebraic $\to$ Trigonometric $\to$ Exponential)
• Standard integral: $\int \cos xdx=\sin x+C$
• Standard integral: $\int \sin xdx=-\cos x+C$
• Basic derivative: $\frac{d}{dx}(x)=1$

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Summary of Steps

1. Identify the integrand as a product requiring integration by parts, choosing $u=x$ and $dv=\cos xdx$ per the LIATE rule.
2. Apply the integration by parts formula: $x\int \cos xdx-\int \left(\frac{d}{dx}(x)\cdot \int \cos xdx\right)dx$.
3. Compute the standard integrals: $\int \cos xdx=\sin x$ and recognize that $\frac{d}{dx}(x)=1$.
4. Simplify the expression to $x\sin x-\int \sin xdx$.
5. Evaluate the remaining integral: $\int \sin xdx=-\cos x$.
6. Combine terms and add the constant of integration: $x\sin x+\cos x+c$.