Assertion (A): if Iₙ = ⁿ x \, dx, then I₆ + I₄ = \; -^5 x/5 Reason (R): ⁿ x \, dx = \; {-ⁿ-1} x}{n} - ⁿ-2} x \, dx

Correct Answer is : A is true, R is false

Correct Answer is : A is true, R is false

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AP EAPCET 04-Jul-2022 Shift 1 Paper

Section: Mathematics

Question : 70 of 160

Marks: +1, -0

Assertion (A): if

I_n = \int \cot^n x \, dx

I_6 + I_4 = \; \frac{-\cot^5 x}{5}

\int \cot^n x \, dx = \; \frac{-\cot^{n-1} x}{n} - \int \cot^{n-2} x \, dx

Solution: 👈: Video Solution

I_n = \int \cot^n x \, dx

I_6 + I_4 \; = \int \cot^6 x \, dx + \int \cot^4 x \, dx

\; = \int \cot^4 x (\cot^2 x + 1) \, dx

\; = -\int \cot^4 x \cdot (-\csc^2 x) \, dx

\; = -\int (\cot x)^4 \cdot (-\csc^2 x) \, dx = - \; \frac{(\cot x)^5}{5}

\therefore

Assertion is true.

\int \cot^n \; x \, dx = \int \cot^{n-2} x \cdot \cot^2 x \, dx

\; = \int \cot^{n-2} x \cdot (\csc^2 x - 1) \, dx

\; = \int \cot^{n-2} x \cdot \csc^2 x \cdot dx - \int \cot^{n-2} x \, dx

\; = -\int (\cot x)^{n-2} x \cdot (-\csc^2 x) \, dx - \int \cot^{n-2} x \, dx

\; = - \; \frac{(\cot x)^{n-1}}{n-1} - \int \cot^{n-2} x \, dx

\therefore

Reason is false.

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