Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

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Let $ℕ$ be the set of natural numbers. For $n \in ℕ,$ define $I_{n} = \int_{0}^{π} \frac{x \sin^{2 n} (x)}{\sin^{2 n} (x) + \cos^{2 n} (x)} d x .$

Then for $m, n \in ℕ$

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Important Questions on Definite Integration

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

\int_{0}^{\infty} e^{- x} \sin^{6} x d x =

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

Let

I_{n} (x) = \int_{0}^{x} \frac{1}{{(t^{2} + 5)}^{n}} d t, n = 1, 2, 3, \dots .

and

{I^{'}}_{n} (x) = \frac{d}{d x} I_{n} (x) .

Then

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

The correct evaluation of

\int_{0}^{π} |\sin^{4} x| d x

is

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

Evaluate :

\int_{0}^{4} {(16 - x^{2})}^{\frac{5}{2}} d x

.

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

If

I_{m, n} = \int_{0}^{1} x^{m - 1} {(1 - x)}^{n - 1} d x

, for

m, n ⩾ 1

, and

\int_{0}^{1} \frac{x^{m - 1} + x^{n - 1}}{{(1 + x)}^{m + n}} d x = α I_{m, n}, α \in R

, then

α

equals ________.

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

Let

I_{n} = \int_{0}^{1} x^{n} \tan^{- 1} x d x .

If

a_{n} I_{n + 2} + b_{n} I_{n} = c_{n}

for all

n \geq 1,

then

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

If

u_{n} = \int_{0}^{\frac{π}{4}} \tan^{n} θ d θ

, then

u_{n} + u_{n - 2}

is

EASY

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

Let

f

and

g

be continuous functions on

[0, a]

such that

f (x) = f (a - x)

and

g (x) + g (a - x) = 4

, then

\int_{0}^{a} f (x) g (x) d x

is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

Given that

\lim_{x \to 0} x^{m} (\log x)^{n} = 0

if

m \geq n

. Let

I_{n} = \int_{0}^{1} x^{n} (\log x)^{n} d x

, then

I_{n}

is equal to

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

If

I_{n} = \int_{0}^{\frac{π}{4}} \tan^{n} x d x

, where

n

is positive integer, then

I_{10} + I_{8}

is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

Find:

\int_{0}^{2 π} \sin^{4} x \cos^{6} x d x

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

Let

f (x) = \{\begin{matrix} \frac{x}{\sin x}, & x \in (0, 1) \\ 1, & x = 0 \end{matrix},

consider the integral

I_{n} = \sqrt{n} \int_{0}^{1 / n} f (x) e^{- n x} d x

. Then

\lim_{n \to \infty} I_{n}

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

For

m, n > 0

, let

α (m, n) = \int_{0}^{2} t^{m} {(1 + 3 t)}^{n} d t

. If ,

11 α (10, 6) + 18 α (11, 5) = p {(14)}^{6}

, then

p

is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

If

\int_{0}^{2 a} x^{2} \sqrt{2 a x - x^{2}} d x = k a^{4}

, then

k : π =

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

The integral

\int_{2}^{4} \frac{log x^{2}}{log x^{2} + log {(6 - x)}^{2}} d x

is equal to

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

The value of the integral

\int_{0}^{\frac{1}{2}} \frac{1 + \sqrt{3}}{{({(x + 1)}^{2} {(1 - x)}^{6})}^{\frac{1}{4}}} d x

is

MEDIUM

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

If

\int_{0}^{π / 3} \frac{\tan θ}{\sqrt{2 k \sec θ}} d θ = 1 - \frac{1}{\sqrt{2}}, (k > 0)

, then the value of

k

is

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

If

b_{n} = \int_{0}^{\frac{π}{2}} \frac{\cos^{2} n x}{\sin x} d x, n \in ℕ

, then

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

If

I_{n} = \int_{\frac{π}{4}}^{\frac{π}{2}} \cot^{n} x d x

, then

HARD

Mathematics>Integral Calculus>Definite Integration>Evaluation of Definite Integral-Different Methods

Let

I_{n} = \int_{0}^{\frac{π}{2}} x^{n} \cos x d x

, where

n

is a non-negative integer. Then

\sum_{n = 2}^{\infty} (\frac{I_{n}}{n!} + \frac{I_{n - 2}}{(n - 2)!})

equals-