Find the Antiderivative g(t)=(6+t+t^2)/( square root of t)

Problem

(∫_^)((6+t+t^2)/√(,t)*d(t))

Solution

1. Rewrite the integrand by expressing the square root as a fractional power and splitting the fraction into three separate terms.

(6+t+t^2)/(t^(1/2))=6/(t^(1/2))+t/(t^(1/2))+(t^2)/(t^(1/2))

2. Simplify each term using the laws of exponents, specifically t^a/t^b=t^(a−b)

6*t^(−1/2)+t^(1/2)+t^(3/2)

3. Apply the power rule for integration, which states (∫_^)(t^n*d(t))=(t^(n+1))/(n+1)+C to each term individually.

(∫_^)((6*t^(−1/2)+t^(1/2)+t^(3/2))*d(t))=6(t^(1/2))/(1/2)+(t^(3/2))/(3/2)+(t^(5/2))/(5/2)+C

4. Simplify the coefficients by multiplying by the reciprocals of the denominators.

12*t^(1/2)+2/3*t^(3/2)+2/5*t^(5/2)+C

Final Answer

(∫_^)((6+t+t^2)/√(,t)*d(t))=12√(,t)+2/3*t^(3/2)+2/5*t^(5/2)+C

Want more problems? Check here!