Question Statement

Evaluate the integral:

$\int \frac{1}{2}\left({\mathrm{cosec}}^{2}x-\mathrm{cosec}x\cot x\right)dx$

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Background and Explanation

This problem requires standard integration formulas for trigonometric functions involving cosecant, along with the linearity properties of integrals (constant multiple rule and difference rule).

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Solution

Let

$I=\int \frac{1}{2}\left({\mathrm{cosec}}^{2}x-\mathrm{cosec}x\cot x\right)dx$

Using the constant multiple rule, we factor out $\frac{1}{2}$, and then apply the difference rule to split the integral:

$=\frac{1}{2}\int {\mathrm{cosec}}^{2}xdx-\frac{1}{2}\int \mathrm{cosec}x\cot xdx$

Now we apply the standard integration formulas. Recall that $\int {\mathrm{cosec}}^{2}xdx=-\cot x$ and $\int \mathrm{cosec}x\cot xdx=-\mathrm{cosec}x$:

$=\frac{1}{2}(-\cot x)-\frac{1}{2}(-\mathrm{cosec}x)+c$

Simplifying the expression by distributing the negative signs:

$=\frac{-1}{2}\cot x+\frac{1}{2}\mathrm{cosec}x+c$

Verification

To confirm our answer is correct, we differentiate the result:

$\frac{d}{dx}\left(\frac{-1}{2}\cot x+\frac{1}{2}\mathrm{cosec}x+c\right)$

$=\frac{-1}{2}\frac{d}{dx}(\cot x)+\frac{1}{2}\frac{d}{dx}(\mathrm{cosec}x)+\frac{d}{dx}(c)$

$=\frac{-1}{2}\left(-{\mathrm{cosec}}^{2}x\right)+\frac{1}{2}(-\mathrm{cosec}x\cot x)+0$

$=\frac{1}{2}{\mathrm{cosec}}^{2}x-\frac{1}{2}\mathrm{cosec}x\cot x$

$=\frac{1}{2}\left({\mathrm{cosec}}^{2}x-\mathrm{cosec}x\cot x\right)$

Since differentiation yields the original integrand, the solution is verified.

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Key Formulas or Methods Used

• $\int {\mathrm{cosec}}^{2}xdx=-\cot x+c$
• $\int \mathrm{cosec}x\cot xdx=-\mathrm{cosec}x+c$
• Constant multiple rule: $\int k\cdot f(x)dx=k\int f(x)dx$
• Difference rule: $\int [f(x)-g(x)]dx=\int f(x)dx-\int g(x)dx$
• Derivative verification: $\frac{d}{dx}(\cot x)=-{\mathrm{cosec}}^{2}x$ and $\frac{d}{dx}(\mathrm{cosec}x)=-\mathrm{cosec}x\cot x$

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Summary of Steps

1. Factor out the constant $\frac{1}{2}$ from the integral using the constant multiple rule
2. Split the integral into two separate integrals using the difference rule
3. Apply the standard integral $\int {\mathrm{cosec}}^{2}xdx=-\cot x$ to the first term
4. Apply the standard integral $\int \mathrm{cosec}x\cot xdx=-\mathrm{cosec}x$ to the second term
5. Simplify the signs to obtain $-\frac{1}{2}\cot x+\frac{1}{2}\mathrm{cosec}x+c$
6. Verify the result by differentiating to recover the original integrand