integrate (t(root t)+t(root t))/t^2 dt
[ t sqrt ( t ) + t sqrt ( t ) ] = 2 t sqrt ( t )
2 t sqrt ( t ) / t ^ 2 = 2 * sqrt ( t ) / [ sqrt ( t ) * sqrt ( t ) * t ] =
2 / sqrt ( t ) = 2 * t ^ ( - 1 / 2 )
int [ 2 * t ^ ( - 1 / 2 ) ] dt =
2 * t ^ ( - 1 / 2 + 1 ) / ( - 1 / 2 + 1 )=
2 * t ^ ( 1 / 2 ) / ( 1 / 2 ) =
2 * 2 * sqrt ( t ) = 4 sqrt ( t ) + C
I think you forgot the extra t in the denominator at
2 / sqrt ( t ) = 2 * t ^ ( - 1 / 2 )
Should be
2t^(-3/2)
and proceed from there...
To integrate the given expression ∫ [(t√t) + (t√t)] / t^2 dt, we can simplify it first.
Step 1: Simplify the expression.
∫ [(t√t) + (t√t)] / t^2 dt
= 2 ∫ [(t√t)] / t^2 dt
Step 2: Simplify the term inside the integral.
2 ∫ [(t√t)] / t^2 dt
= 2 ∫ (√t) / t dt
Step 3: Rewrite the expression using exponentials.
2 ∫ t^(-1/2) dt
Step 4: Apply the power rule of integration.
2 ∫ t^(-1/2) dt
= 2 * (t^(1/2)) / (1/2) + C
= 4√t + C
So, the integral of the given expression is 4√t + C, where C is the constant of integration.
To integrate the given expression ∫ (t(√t) + t(√t)) / t^2 dt, we can simplify it by combining similar terms.
Step 1: Simplify the expression
The given expression can be written as ∫ (2t^(3/2)) / t^2 dt.
Step 2: Combine like terms
We can simplify further by canceling out one of the t terms in the denominator. Rewrite the expression as ∫ 2t^(3/2) / t^2 dt.
Step 3: Divide the exponents
Since we have the same base (t), we can divide the exponents and subtract them. t^(3/2) / t^2 can be simplified as t^((3/2) - 2) = t^(-1/2).
Step 4: Integrate the expression
Now, the integral can be written as ∫ 2t^(-1/2) dt.
To integrate this, we use the power rule of integration, which states that the integral of t^n dt = (t^(n+1)) / (n+1).
Applying the power rule, the integral becomes 2 * (t^(-1/2 + 1)) / (-1/2 + 1) = -4t^(1/2) / (1/2) + C.
Simplifying further, we get -8t^(1/2) + C, where C is the constant of integration.