Calculus

Integrals

Medium

applied

$\int \frac{dx}{x(x^{10}+1)}$ is equal to

$\log\left|\frac{x^{10}}{x^{10}+1}\right|+C$ , where C is an arbitrary constant

$\log\left|\frac{x^{10}+1}{x^{10}}\right|+C$ , where C is an arbitrary constant

$\frac{1}{10}\log\left|\frac{x^{10}+1}{x^{10}}\right|+C$ , where C is an arbitrary constant

$\frac{1}{10}\log\left|\frac{x^{10}}{x^{10}+1}\right|+C$ , where C is an arbitrary constant

✅ Correct Option: 4

Multiply numerator and denominator by $x^9$ and put $u=x^{10}$ , $du=10x^9dx$ . The integral becomes $\frac{1}{10}\int\frac{du}{u(u+1)}=\frac{1}{10}\log\left|\frac{u}{u+1}\right|$ , i.e. $\frac{1}{10}\log\left|\frac{x^{10}}{x^{10}+1}\right|+C$ .

2025: 26 May Shift 2

The value of $\int_{0}^{\pi/2} \frac{\tan^7 x}{\cot^7 x + \tan^7 x} dx$ is

2026: 26 May Shift 2

The value of $\displaystyle\int_0^1 xe^{x^2}\,dx$ is equal to :

2026: 21 May Shift 1

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

2022: 10 Aug Shift 1

Match List I with List II

List - I	List - II
A. $\int_{-\pi/2}^{\pi/2} \sin^7 x \, dx$	I. $\frac{\pi}{2}$
B. $\int_{-\pi/2}^{\pi/2} \sin^2 x \, dx$	II. $\frac{\pi}{4}$
C. $\int_0^{\pi/2} \frac{1}{1 + \tan x} \, dx$	III. 0
D. $\int_0^{\pi} \lvert \cos x \rvert \, dx$	IV. 2

2026: 25 May Shift 1

$\displaystyle\int_{-\pi/3}^{\pi/3} 2x^5\cos^4 x\, dx$ equals

2025: 27 May Shift 1

The value of $\int \frac{x^5}{\sqrt{1 + x^3}} dx$ is

∫dxx(x10+1)\int \frac{dx}{x(x^{10}+1)}∫x(x10+1)dx​ is equal to

Solution

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$\int \frac{dx}{x(x^{10}+1)}$ is equal to