Evaluate the following indefinite integrals using substitution
a) ∫ xsqrt(x^2-7) dx
b) ∫ x^(2/3)(1/5x^(5/3)+2)^4 dx
(a) u = x^2-7
(b) u = 1/5 x^(5/3) + 2
To evaluate the given indefinite integrals using substitution, we need to choose an appropriate substitution that simplifies the integral. Let's go through each of the integrals step by step.
a) ∫ xsqrt(x^2-7) dx:
Step 1: Choose the substitution.
Let's substitute u = x^2-7. Then, we can express the integral in terms of u: xsqrt(u) dx.
Step 2: Find the derivative of the substitution.
To find dx in terms of du, we differentiate both sides of the substitution equation.
du = 2x dx, which implies dx = du/(2x).
Step 3: Substitute the variables and simplify the integral.
The integral becomes:
∫ xsqrt(x^2-7) dx = ∫ xsqrt(u) dx = ∫ xsqrt(u) (du/(2x)) = 1/2 ∫ sqrt(u) du.
Step 4: Evaluate the new integral.
∫ sqrt(u) du can be evaluated as (2/3)u^(3/2) + C, where C is the constant of integration.
Step 5: Substitute back the original variable.
Since our original substitution was u = x^2-7, we substitute back u to get the final result:
∫ xsqrt(x^2-7) dx = 1/2 ∫ sqrt(u) du = 1/2 ((2/3)(x^2-7)^(3/2)) + C = (x^2-7)^(3/2)/3 + C.
Therefore, the solution to the integral ∫ xsqrt(x^2-7) dx is (x^2-7)^(3/2)/3 + C, where C is the constant of integration.
b) ∫ x^(2/3)(1/5x^(5/3)+2)^4 dx:
Step 1: Choose the substitution.
Let's substitute u = 1/5x^(5/3) + 2. Then, we can express the integral in terms of u: x^(2/3)(u)^4 dx.
Step 2: Find the derivative of the substitution.
To find dx in terms of du, we differentiate both sides of the substitution equation.
du = (2/3)x^(-1/3)(1/5) dx, which implies dx = (3/2)x^(1/3) du.
Step 3: Substitute the variables and simplify the integral.
The integral becomes:
∫ x^(2/3)(1/5x^(5/3)+2)^4 dx = ∫ x^(2/3)(u)^4 (3/2)x^(1/3) du = (3/2) ∫ (x^((2/3)+(1/3))) (u)^4 du = (3/2) ∫ x (u)^4 du.
Step 4: Evaluate the new integral.
∫ x (u)^4 du can be evaluated as (∫ (u)^4 dx) * x^2/2 + C, where C is the constant of integration.
Step 5: Substitute back the original variable.
Since our original substitution was u = 1/5x^(5/3) + 2, we substitute back u to get the final result:
∫ x^(2/3)(1/5x^(5/3)+2)^4 dx = (3/2) (∫ (u)^4 dx) * x^2/2 + C = (3/2) ((1/5)^4 (x^(5/3))^4) * x^2/2 + C = (3/2) (1/625) (x^(4/3)) * x^2/2 + C.
Simplifying further, we get:
∫ x^(2/3)(1/5x^(5/3)+2)^4 dx = (3/2) (1/625) (x^(4/3)) * x^2/2 + C = 3x^8/625 + C.
Therefore, the solution to the integral ∫ x^(2/3)(1/5x^(5/3)+2)^4 dx is 3x^8/625 + C, where C is the constant of integration.