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Visual Solutions

11 step-by-step animated solutions for Definite & Indefinite Integrals. Each solution breaks the problem into visual clips with key insights.

PYQ 2024 · Jan Shift 2Standard4 steps
The value of the integral ∫0π/4x dxsin⁡4(2x)+cos⁡4(2x)\int_0^{\pi/4} \frac{x \, dx}{\sin^4(2x) + \cos^4(2x)} equals:
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PYQ 2024 · Jan Shift 2Easy4 steps
The value of ∫01(2x3−3x2−x+1)1/3 dx\int_0^1 (2x^3 - 3x^2 - x + 1)^{1/3} \, dx is equal to:
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PYQ 2024 · Jan Shift 2Standard4 steps
If ∫0π/3cos⁡4x dx=aπ+b3\int_0^{\pi/3} \cos^4 x \, dx = a\pi + b\sqrt{3}, where aa and bb are rational numbers, then 9a+8b9a + 8b is equal to:
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PYQ 2024 · Jan Shift 2Standard4 steps
For 0<a<10 < a < 1, the value of the integral ∫0πdx1−2acos⁡x+a2\int_0^{\pi} \frac{dx}{1 - 2a\cos x + a^2} is:
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PYQ 2024 · Jan Shift 2Tricky4 steps
Let f:(0,∞)→Rf : (0,\infty) \to \mathbb{R} and F(x)=∫0xtf(t) dtF(x) = \int_0^x t f(t) \, dt. If F(x2)=x4+x5F(x^2) = x^4 + x^5, then ∑r=112f(r2)\sum_{r=1}^{12} f(r^2) is equal to:
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PYQ 2024 · Apr Shift 1Standard4 steps
The value of ∫−ππ2y(1+sin⁡y)1+cos⁡2y dy\int_{-\pi}^{\pi} \frac{2y(1+\sin y)}{1+\cos^2 y} \, dy is:
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PYQ 2024 · Apr Shift 1Standard4 steps
The integral ∫0π/4cos⁡2xsin⁡2x(cos⁡3x+sin⁡3x)2 dx\int_0^{\pi/4} \frac{\cos^2 x \sin^2 x}{(\cos^3 x + \sin^3 x)^2} \, dx is equal to:
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PYQ 2024 · Jan Shift 1Standard4 steps
The integral ∫(x8−x2) dx(x12+3x6+1)tan⁡−1(x3+1x3)\int \frac{(x^8 - x^2) \, dx}{(x^{12} + 3x^6 + 1) \tan^{-1}\left(x^3 + \frac{1}{x^3}\right)} equal to:
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PYQ 2024 · Apr Shift 2Standard4 steps
If the value of the integral ∫−11cos⁡αx1+3x dx\int_{-1}^{1} \frac{\cos \alpha x}{1+3^x} \, dx is 2π\frac{2}{\pi}, then a value of α\alpha is:
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PYQ 2024 · Apr Shift 1Tricky4 steps
Let I(x)=∫6sin⁡2x(1−cot⁡x)2 dxI(x) = \int \frac{6}{\sin^2 x(1-\cot x)^2} \, dx. If I(0)=3I(0) = 3, then I(π12)I\left(\frac{\pi}{12}\right) is equal to:
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PYQ 2023 · Apr Shift 1Standard4 steps
The value of ∫0π/2sin⁡2x1+sin⁡xcos⁡x dx\int_0^{\pi/2} \frac{\sin^2 x}{1 + \sin x \cos x} \, dx is:
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