Question

integration upper limit 2 lower limit minus 2 (x power 2019) dx is equal to​

Answer 1

Step-by-step explanation:

We know that

∫

a

b

f(x)dx=(b−a)

n→∞

lim

n

1

(f(a)+f(a+h)+...+f(a+(n−1)h))

Putting a=0,b=2,h=

n

b−a

=

n

2−0

=

n

2

in ∫

0

2

x

2

+1dx

I=(2−0)

n→∞

lim

n

1

(f(0)+f(n)+f(2n)+...+f((n−1)h))

f(0)=1

f(h)=h

2

+1

=(

n

2

4

)+1

f((n−1)h)=(n−1)

2

×

n

2

4

+1

∴I=2

n→∞

lim

n

1

((1+1+...ntimes)+(0+

n

2

4

+

n

2

16

+...+

n

2

(n−1)

2

))

=2

n→∞

lim

n

1

(n+

n

4

6

(n−1)n(2n−1)

)

=2

n→∞

lim

(1+

3

2

(1−

n

1

)(2−

n

1

))

=2×(1+

3

4

)=

3

14