1
AP EAPCET 2022 - 4th July Morning Shift
+1
-0

Assertion (A) If $$I_n=\int \cot ^n x d x$$, then $$I_6+I_4=\frac{-\cot ^5 x}{5}$$

Reason (R) $$\int \cot ^n x d x=\frac{-\cot ^{n-1} x}{n} -\int \cot ^{n-2} x d x$$

A
A is false, R is false
B
A is true, R is true
C
A is true, R is false
D
A is false, R is true
2
AP EAPCET 2022 - 4th July Morning Shift
+1
-0

If $$I_n=\int \tan ^n x d x$$, and $$I_0+I_1+2 I_2+2 I_3+2 I_4 +I_5+I_6=\sum_\limits{k=1}^n \frac{\tan ^k x}{k}$$, then $$n=$$

A
6
B
5
C
4
D
3
3
AP EAPCET 2022 - 4th July Morning Shift
+1
-0

$$\int \frac{e^{\cot x}}{\sin ^2 x}(2 \log \operatorname{cosec} x+\sin 2 x) d x=$$

A
$$-2 e^{\cot x} \log \left(\operatorname{cosec}^2 x\right)+C$$
B
$$-2 e^{\cot x} \log (\operatorname{cosec} x)+C$$
C
$$-2 e^{\cot x} \log (\operatorname{cosec} x+\sin x)+C$$
D
$$-2 e^{\cot x} \log (\operatorname{cosec} x-\cot x)+C$$
4
AP EAPCET 2022 - 4th July Morning Shift
+1
-0

The parametric form of a curve is $$x=\frac{t^3}{t^2-1} y=\frac{t}{t^2-1}$$, then $$\int \frac{d x}{x-3 y}=$$

A
$$\frac{1}{2} \log \left(t^2-1\right)+C$$
B
$$2 \log \left(t\left(t^2-1\right)\right)+C$$
C
$$\frac{1}{4} \log \left(\frac{t}{t^2-3}\right)+C$$
D
$$\frac{5}{2} \log \left(t+\frac{1}{t^2}\right)+C$$
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