Grade 10Integral Calculus

options

3/π log2

2/πlog2

3/π log3

3/π

Aditya Kartikeya

12 Years agoGrade 10

Jitender Singh

12 Years ago

Ans:
Hello Student,
Please find answer to your question below

$L = \lim_{n\rightarrow \infty }[tan\frac{\pi }{3n}+tan\frac{2\pi }{3n}+tan\frac{3\pi }{3n}+.................+tan\frac{\pi }{3}]\frac{1}{n}$
$L = \lim_{n\rightarrow \infty }[\sum_{r = 1}^{n}tan\frac{r\pi }{3n}]\frac{1}{n}$
$L = \lim_{n\rightarrow \infty }[\sum_{r = 1}^{n}\frac{1}{n}.tan\frac{r\pi }{3n}]$
Now we are going to convert it into definite integral using the definition
$\frac{r}{n} = x,\sum_{r = 1}^{n}\frac{1}{n} = dx$
Limit is from 0 to 1.
$L = \int_{0}^{1}tan\frac{\pi x}{3}dx$
$L = \int_{0}^{1}\frac{sin\frac{\pi x}{3}}{cos\frac{\pi x}{3}}dx$
$L = \frac{3}{\pi }\int_{0}^{1}\frac{\frac{\pi }{3}.sin\frac{\pi x}{3}}{cos\frac{\pi x}{3}}dx$
$cos\frac{\pi x}{3} = t$
$-\frac{\pi }{3}sin\frac{\pi x}{3}dx = dt$
$x = 0\rightarrow t = 1$
$x = 1\rightarrow t = \frac{1}{2}$
$L = \frac{3}{\pi }\int_{1}^{1/2}\frac{-1}{t}dt$
$L = \frac{3}{\pi }\int_{1/2}^{1}\frac{1}{t}dt$
$L = \frac{3}{\pi }[logt]_{1/2}^{1}$
$L = \frac{3}{\pi }[-log(1/2)]$
$L = \frac{3}{\pi }[log(2)]$
Option (1) is correct.

options

3/π log2

2/πlog2

3/π log3

3/π

Aditya Kartikeya

1 Answer

Jitender Singh

The area of a square field is 640000cm².what is its area in hectares

Please solve this integration. I am facing difficulty in it
How to solve this. Please solve anyone. Humble request

Using laplace transform, solve d²y/dt²+ dy/dt = t² +2t given that y=4 and y’=-2 and t=0

If 4th term of an A.p is zero, show that the 8th term is double the 6th term

options 3/π log2 2/πlog2 3/π log3 3/π

Aditya Kartikeya

1 Answer

Jitender Singh

Other Related Questions on integral calculus

The area of a square field is 640000cm2.what is its area in hectares

Please solve this integration. I am facing difficulty in it How to solve this. Please solve anyone. Humble request

Using laplace transform, solve d2y/dt2 + dy/dt = t2 +2t given that y=4 and y’=-2 and t=0

If 4th term of an A.p is zero, show that the 8th term is double the 6th term

options

3/π log2

2/πlog2

3/π log3

3/π

The area of a square field is 640000cm².what is its area in hectares

Please solve this integration. I am facing difficulty in it
How to solve this. Please solve anyone. Humble request

Using laplace transform, solve d²y/dt²+ dy/dt = t² +2t given that y=4 and y’=-2 and t=0