Answer to Question #98133 in Calculus for Frax

Question

Evaluate the following integrals. Show complete solution.

1. Integral of 3dy/ y raised to the fourth power
2. Integral of square root of x minus x/3 plus 2/ square root of x dx

Answer 1

The solution of the problem uses a general formula for the integral of the power function

$\int x^n dx=x^{n+1}/(n+1)$

1 Evaluate the integral$\int 3dy/y^4$

Solution: $\int 3dy/y^4=3\int y^{-4}dy=3 y^{-4+1}/(-4+1)+C=3 y^{-3}/(-3)+C=-1/y^3+C$

Answer: $\int 3dy/y^4=-1/y^3+C$

2 Evaluate the integral $\int (\sqrt{x}-x/3+2/\sqrt{x})dx$

Solution: $\int (\sqrt{x}-x/3+2/\sqrt{x})dx=I_1- (1/3) I_2+2 I_3$

$I_1=\int\sqrt{x}dx= \int x^{1/2} dx =x^{1/2+1}/(1/2+1)+C_1= x \sqrt{x}/(3/2)+C_1=2x\sqrt{x}/3+C_1$

$I_2=\int x dx=x^{1+1}/(1+1)+C_2=x^2/2+C_2$

$I_3=\int dx/\sqrt{x}=\int x^{-1/2} dx=x^{1-1/2}/(1-1/2)+C_3=x^{1/2}/(1/2)+C_3=2\sqrt{x}+C_3$

Substitution integrals into the formula gives the answer.

Answer: $\int (\sqrt{x}-x/3+2/\sqrt{x})dx=2 x \sqrt{x}/3 - x^2/6 + 4 \sqrt{x}+C$