If (^6 x)/(^10 x) dx = -2a^9 x + b^7 x + c, (where c is an arbitrary constant), then which one of the following is TRUE? Option 1: 18a=7b Option 2: 18a+7b=0 Option 3: 7a=18b Option 4: 7a+18b=0

Answer: 18a+7b=0. Solution: Write the integrand as ^6x^4x=^6x(1+^2x)^2x. With t= x, the integral is t^7/7+t^9/9. Comparing, -2a=19 and b=17, so 18a=-1 and 7b=1, giving 18a+7b=0.

Calculus

Integrals

Hard

core

If $\int \frac{\sin^6 x}{\cos^{10} x}\,dx = -2a\tan^9 x + b\tan^7 x + c$ , (where c is an arbitrary constant), then which one of the following is TRUE?

18a=7b

18a+7b=0

7a=18b

7a+18b=0

✅ Correct Option: 2

Write the integrand as $\tan^6x\sec^4x=\tan^6x(1+\tan^2x)\sec^2x$ . With $t=\tan x$ , the integral is $\frac{t^7}{7}+\frac{t^9}{9}$ . Comparing, $-2a=\frac19$ and $b=\frac17$ , so $18a=-1$ and $7b=1$ , giving $18a+7b=0$ .

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The value of $\int_{-2}^{3} |x-1|\, dx$ is

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$\int_0^{\pi/2} \frac{\sin^8 x}{\sin^8 x + \cos^8 x} dx$ is equal to

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The value of $\int \left(\sqrt{x} + \frac{1}{\sqrt{x}}\right)^2 dx$ is:

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Match List-I with List-II

List-I	List-II
Indefinite integral	Solution (where 'c' is an arbitrary constant)
(A) $\displaystyle\int \sqrt{x^2-a^2} \, dx$	(I) $\dfrac{1}{a}\tan^{-1}\dfrac{x}{a} + c$
(B) $\displaystyle\int \dfrac{1}{x^2-a^2} \, dx$	(II) $\dfrac{x}{2}\sqrt{x^2+a^2} + \dfrac{a^2}{2}\log\left[x+\sqrt{x^2+a^2}\right] + c$
(C) $\displaystyle\int \dfrac{1}{x^2+a^2} \, dx$	(III) $\dfrac{x}{2}\sqrt{x^2-a^2} - \dfrac{a^2}{2}\log\left[x+\sqrt{x^2-a^2}\right] + c$
(D) $\displaystyle\int \sqrt{x^2+a^2} \, dx$	(IV) $\dfrac{1}{2a}\log\left\vert \dfrac{x-a}{x+a} \right\vert + C$

Choose the correct answer from the options given below:

2023: 22 May Shift 3

$\int_0^{\pi/2} \sqrt{1 - \sin 2x} \, dx$ is equal to :

If ∫sin⁡6xcos⁡10x dx=−2atan⁡9x+btan⁡7x+c\int \frac{\sin^6 x}{\cos^{10} x}\,dx = -2a\tan^9 x + b\tan^7 x + c∫cos10xsin6x​dx=−2atan9x+btan7x+c, (where c is an arbitrary constant), then which one of the following is TRUE?

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CUET Mathematics Past Year Questions (Topic-Wise)

If $\int \frac{\sin^6 x}{\cos^{10} x}\,dx = -2a\tan^9 x + b\tan^7 x + c$ , (where c is an arbitrary constant), then which one of the following is TRUE?