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Important Questions: Fundamental integrals involving algebraic, trigonometric, exponential and logarithmic functions for JEE

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Part of Integral Calculus in JEE Main Mathematics.

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Q1MathsUnit 8: Integral Calculus
∫(e5log⁡x−e4log⁡xe3log⁡x−e2log⁡x)dx=\int\left(\frac{e^{5 \log x}-e^{4 \log x}}{e^{3 \log x}-e^{2 \log x}}\right) d x=
Q2MathsUnit 8: Integral Calculus
STATEMENT - 1: The volume of largest sphere that can be carved out from cube of side a cm is 16πa3\frac{1}{6} \pi a^{3} STATEMENT - 2: Volume of sphere is 43πr3\frac{4}{3} \pi r^{3} and for largest sphere to carved from cube radius of sphere == side of cube
Q3MathsUnit 8: Integral Calculus
A hollow spherical shell is made of metal of density 4.8g/cm3.4.8 \mathrm{g} / \mathrm{cm}^{3} . If its internal and external radii are 10cm10 \mathrm{cm} and 12cm12 \mathrm{cm} respectively, find the weight of the shell
Q4MathsUnit 8: Integral Calculus
Assertion If a>0a>0 and b2−4ac<0.b^{2}-4 a c<0 . then the value of the integral ∫dxax2+bx+c\int \frac{d x}{a x^{2}+b x+c} will be of the type μtan⁡−1(x+AB)+\mu \tan ^{-1}\left(\frac{x+A}{B}\right)+ C;C ; where A,B,C,μA, B, C, \mu are constant. Reason f(a>0,b2−4ac<0, then ax2+bx+f\left(a>0, b^{2}-4 a c<0, \text { then } a x^{2}+b x+\right. cc can be written as sum of two squares.

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