Mathematics Crash Course JEE Main>Chapter 7 - Definite Integration>Level 3>Q 5

MEDIUM

JEE Main

IMPORTANT

Earn 100

The value of $\frac{(5050) \int_{0}^{1} {(1 - x^{50})}^{100} d x}{\int_{0}^{1} {(1 - x^{50})}^{101} d x}$ is

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Important Questions on Definite Integration

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 7 - Definite Integration>Level 3>Q 6

Let

a = \int_{0}^{\log 2} \frac{2 e^{3 x} + e^{2 x} - 1}{e^{3 x} + e^{2 x} - e^{x} + 1} d x,

then

4 e^{a} =

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 7 - Definite Integration>Level 3>Q 7

If

a > 0

and

A = \int_{0}^{a} \cos^{- 1} x d x,

then

\int_{- a}^{a} (\cos^{- 1} x - \sin^{- 1} \sqrt{1 - x^{2}}) d x = π a - λ A,

then

λ

is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 7 - Definite Integration>Level 3>Q 8

\int_{\frac{π}{4}}^{\frac{3 π}{4}} \frac{x}{1 + \sin x} d x =

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 7 - Definite Integration>Level 3>Q 9

The value of

\int_{\sqrt{\log_{e} 2}}^{\sqrt{\log_{e} 3}} \frac{x \sin x^{2}}{\sin x^{2} + \sin (\log_{e} 6 - x^{2})} d x

is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 7 - Definite Integration>Level 3>Q 10

\int_{3}^{4} \sqrt{\frac{x - 3}{4 - x}} d x =

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 7 - Definite Integration>Level 3>Q 11

\int_{- π}^{π} \frac{2 x (1 + \sin x)}{1 + \cos^{2} x} d x

is

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 7 - Definite Integration>Level 3>Q 12

The value of $f (k) = \int_{0}^{\frac{π}{2}} \log (\sin^{2} θ + k^{2} \cos^{2} θ) d θ$ is equal to

HARD

JEE Main

IMPORTANT

Mathematics Crash Course JEE Main>Chapter 7 - Definite Integration>Level 3>Q 13

If

f (x) = \frac{1}{2^{n}}

when

\frac{1}{2^{n + 1}} < x \leq \frac{1}{2^{n}}, n = 0, 1, 2, \dots,

then

\lim_{n \to \infty} \int_{\frac{1}{2^{n}}}^{1} f (x) d x

equals to