Question Statement

Evaluate the integral: $\int x{\sec }^{2}xdx$

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Background and Explanation

This problem requires integration by parts, a technique for integrating products of functions. Following the LIATE rule (Logarithmic, Inverse trig, Algebraic, Trigonometric, Exponential), we select the algebraic function $x$ as $u$ and the trigonometric function ${\sec }^{2}xdx$ as $dv$.

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Solution

Let: $I=\int x{\sec }^{2}xdx$

Step 1: Apply integration by parts using the formula $\int udv=uv-\int vdu$.

Identify:

• $u=x\Rightarrow \frac{d}{dx}(x)=1$
• $dv={\sec }^{2}xdx\Rightarrow v=\int {\sec }^{2}xdx=\tan x$

Writing out the application: $I=x\cdot \int {\sec }^{2}xdx-\int \left(\int {\sec }^{2}xdx\right)\cdot \frac{d}{dx}(x)dx$

Step 2: Substitute $v=\tan x$ and simplify: $=x\cdot \tan x-\int \tan x\cdot 1dx$

Step 3: Simplify the expression: $=x\tan x-\int \tan xdx$

Step 4: Rewrite $\tan x$ as $\frac{\sin x}{\cos x}$: $=x\tan x-\int \frac{\sin x}{\cos x}dx$

Step 5: Multiply numerator and denominator by $-1$ to prepare for integration (recognizing that the derivative of $\cos x$ is $-\sin x$): $=x\tan x+\int \frac{-\sin x}{\cos x}dx$

Step 6: Integrate using the rule $\int \frac{f^{\prime }(x)}{f(x)}dx=\ln |f(x)|+C$: $=x\tan x+\ln (\cos x)+c$

Step 7: Using the identity $\ln (\cos x)=-\ln (\sec x)$, we can also express the answer as: $=x\tan x-\ln (\sec x)+c$

Therefore, the solution is: $\boxed{x\tan x+\ln (\cos x)+c}\text{or equivalently}\boxed{x\tan x-\ln (\sec x)+c}$

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Key Formulas or Methods Used

• Integration by parts: $\int udv=uv-\int vdu$
• Standard integral: $\int {\sec }^{2}xdx=\tan x+C$
• Trigonometric identity: $\tan x=\frac{\sin x}{\cos x}$
• Logarithmic integration: $\int \frac{f^{\prime }(x)}{f(x)}dx=\ln |f(x)|+C$ (where $f(x)=\cos x$ and $f^{\prime }(x)=-\sin x$)
• Logarithm property: $\ln (\cos x)=-\ln (\sec x)$

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Summary of Steps

1. Set up: Define $I=\int x{\sec }^{2}xdx$ and choose $u=x$, $dv={\sec }^{2}xdx$ for integration by parts
2. Compute parts: Determine $du=dx$ and $v=\tan x$
3. Apply formula: Obtain $x\tan x-\int \tan xdx$
4. Rewrite: Express $\tan x=\frac{\sin x}{\cos x}$
5. Adjust signs: Rewrite as $x\tan x+\int \frac{-\sin x}{\cos x}dx$ to match derivative pattern
6. Integrate: Evaluate to get $\ln (\cos x)$
7. Final answer: Combine terms and present alternative form $x\tan x-\ln (\sec x)+c$