{"id":9001,"date":"2024-02-04T18:10:28","date_gmt":"2024-02-04T12:40:28","guid":{"rendered":"https:\/\/physicscatalyst.com\/article\/?p=9001"},"modified":"2024-02-04T18:10:32","modified_gmt":"2024-02-04T12:40:32","slug":"integration-of-cot-square-x","status":"publish","type":"post","link":"https:\/\/physicscatalyst.com\/article\/integration-of-cot-square-x\/","title":{"rendered":"integration of cot square x"},"content":{"rendered":"\n

The integration of cot square x $\\cot ^2 x$, can be found using trigonometry identities and fundamental integral formulas . The integral of $\\cot^2 x$ with respect to (x) is:<\/p>\n\n\n\n

\\[
\\int \\cot^2 x \\, dx =-\\cot x -x + C
\\]<\/p>\n\n\n\n

Here, (C) represents the constant of integration, which is added because the process of integration determines the antiderivative up to an arbitrary constant.<\/p>\n\n\n\n

Proof of integration of cot square x<\/h2>\n\n\n\n

We know from trigonometry identities
$cosec^2 x =1 + \\cot^2 x$
or $\\cot^2 x =\\csc^2 x -1$
$\\int \\cot^2 x dx = \\int (\\csc^2 x -1) dx$
Now we know that from fundamental integration formula
$\\int ( \\csc^2 x) \\; dx = -\\cot x + C$
Therefore
$$\\int \\cot^2 x \\; dx = \\int (\\csc^2 x -1) dx= -\\cot x -x + C$<\/p>\n\n\n\n

Definite Integral of cot square x<\/h2>\n\n\n\n

To find the definite integral of $\\cot^2x$ over a specific interval, we use the same approach as with the indefinite integral, but we’ll apply the limits of integration at the end.<\/p>\n\n\n\n

The definite integral of $\\cot^2x$ from $a$ to $b$ is given by:<\/p>\n\n\n\n

$$\\int_{a}^{b} \\cot^2x \\, dx = \\cot (a) – \\cot (b) + a -b $$<\/p>\n\n\n\n

This expression represents the accumulated area under the curve of $\\cot^2x$ from $x = a$ to $x = b$.<\/p>\n\n\n\n

Solved Examples<\/h2>\n\n\n\n

Question 1<\/strong><\/p>\n\n\n\n

$$ \\int x \\cot^2 x \\; dx $$<\/p>\n\n\n\n

Solution<\/strong><\/p>\n\n\n\n

Using Integration by Parts,<\/p>\n\n\n\n