Mathematics · JEE

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

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Q1MathsUnit 8: Integral Calculus

\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

\frac{1}{6} \pi a^{3}

STATEMENT - 2: Volume of sphere is

\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

4.8 \mathrm{g} / \mathrm{cm}^{3} .

If its internal and external radii are

10 \mathrm{cm}

and

12 \mathrm{cm}

respectively, find the weight of the shell

Q4MathsUnit 8: Integral Calculus

Assertion If

a>0

and

b^{2}-4 a c<0 .

then the value of the integral

\int \frac{d x}{a x^{2}+b x+c}

will be of the type

\mu \tan ^{-1}\left(\frac{x+A}{B}\right)+

C ;

where

A, B, C, \mu

are constant. Reason

f\left(a>0, b^{2}-4 a c<0, \text { then } a x^{2}+b x+\right.

c

can be written as sum of two squares.

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