Answer to Question #98133 in Calculus for Frax

Question #98133
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
1
Expert's answer
2019-11-12T11:39:57-0500

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"


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