Use the substitution x=2tan(theta) to evaluate the indefinite integral 53dx/(x^2sqrt(x2+4))
To evaluate the indefinite integral ∫ (53dx) / (x^2*sqrt(x^2+4)), let's use the substitution x = 2tan(θ).
First, let's find the derivative of x in terms of θ:
dx = 2sec^2(θ)dθ
Now, substitute these expressions for x and dx into the integral:
∫ (53dx) / (x^2*sqrt(x^2+4)) = ∫ (53(2sec^2(θ)dθ)) / ((2tan^2(θ))^2*sqrt((2tan(θ))^2+4))
Simplifying further, we have:
∫ (106sec^2(θ)dθ) / (4tan^4(θ)*sqrt(4tan^2(θ)+4))
= ∫ (53sec^2(θ)dθ) / (2tan^4(θ)*sqrt(tan^2(θ)+1))
Since tan^2(θ)+1 = sec^2(θ), we can simplify it further to:
∫ (53sec^2(θ)dθ) / (2tan^4(θ)*sqrt(sec^2(θ)))
= ∫ (53sec^2(θ)dθ) / (2tan^4(θ)*sec(θ))
= 53/2 ∫ sec(θ)/tan^4(θ) dθ
Now, let's convert the remaining trigonometric functions into terms of sin and cos:
sec(θ) = 1/cos(θ)
tan(θ) = sin(θ)/cos(θ)
Substituting these expressions into the integral, we get:
= 53/2 ∫ (1/cos(θ)) / ((sin(θ)/cos^4(θ))) dθ
= 53/2 ∫ (cos^3(θ)) / sin(θ) dθ
Now, let's make a substitution to simplify the integral further. We'll use u = sin(θ).
Then, du = cos(θ) dθ.
Substituting these into the integral, we have:
= 53/2 ∫ u^3 du
= 53/2 * (u^4 / 4) + C
where C is the constant of integration.
Now, substitute back u = sin(θ):
= 53/2 * (sin^4(θ) / 4) + C
Finally, substitute back x = 2tan(θ):
= 53/8 * (2tan^4(θ) / 4) + C
= 53/16 * tan^4(θ) + C
Therefore, the indefinite integral ∫ (53dx) / (x^2*sqrt(x^2+4)) simplifies to:
53/16 * tan^4(θ) + C, where θ is the angle corresponding to the substitution x = 2tan(θ), and C is the constant of integration.
To evaluate the indefinite integral ∫ (53dx)/(x^2√(x^2+4)) using the substitution x = 2tan(θ), we need to follow these steps:
Step 1: Find dx in terms of dθ.
Taking the derivative of both sides with respect to θ:
dx = 2sec^2(θ)dθ
Step 2: Substitute x = 2tan(θ) and dx = 2sec^2(θ)dθ into the integral.
∫ (53dx)/(x^2√(x^2+4)) = ∫ (53*2sec^2(θ)dθ)/[(2tan(θ))^2√((2tan(θ))^2+4)]
Simplifying the expression:
∫ (53*2sec^2(θ)dθ) / (4tan^2(θ)√(4tan^2(θ)+4))
= ∫ (106sec^2(θ)dθ) / (4tan^2(θ)√(4(tan^2(θ)+1)))
= ∫ (53sec^2(θ)dθ) / (tan^2(θ)√(2(tan^2(θ)+1)))
Step 3: Simplify the integrand using trigonometric identities.
Using the identity sec^2(θ) = 1 + tan^2(θ):
= ∫ (53(dθ))/(tan^2(θ)√(2(tan^2(θ)+1)))
= ∫ (53(dθ))/(tan^2(θ)√(2sec^2(θ)))
= ∫ (53(dθ))/sqrt(2)sec(θ)tan^2(θ)
Step 4: Convert the remaining trigonometric functions in terms of one variable.
Using the identity sec(θ) = sqrt(1 + tan^2(θ)):
= ∫ (53(dθ))/(sqrt(2)(1 + tan^2(θ))tan^2(θ))
= ∫ (53(dθ))/(sqrt(2)(tan^2(θ) + 1)tan^2(θ))
= ∫ (53(dθ))/(sqrt(2)tan^2(θ)(tan^2(θ) + 1))
Step 5: Convert the integrand into a rational function.
Using the identity tan^2(θ) = sec^2(θ) - 1:
= ∫ (53(dθ))/(sqrt(2)tan^2(θ)((sec^2(θ) - 1) + 1))
= ∫ (53(dθ))/(sqrt(2)tan^2(θ)(sec^2(θ)))
= ∫ (53(dθ))/(sqrt(2)sec^2(θ))
= ∫ (53(dθ))/(sqrt(2)sec^2(θ))
Step 6: Simplify and integrate.
= (53/sqrt(2)) ∫ (dθ)/sec^2(θ)
= (53/sqrt(2)) ∫ cos^2(θ)dθ [Using the identity sec^2(θ) = 1/cos^2(θ)]
Now, the integral becomes easier to evaluate:
= (53/sqrt(2)) ∫ [1 - sin^2(θ)]dθ
= (53/sqrt(2)) [θ - (1/3)sin^3(θ)] + C [Integrating term-by-term]
Step 7: Convert back to the original variable x.
= (53/sqrt(2)) [θ - (1/3)sin^3(θ)] + C
= (53/sqrt(2)) [arctan(x/2) - (1/3)(x/2)√(4 + x^2)] + C
Therefore, the indefinite integral ∫ (53dx)/(x^2√(x^2+4)) is equal to (53/sqrt(2)) [arctan(x/2) - (1/3)(x/2)√(4 + x^2)] + C.
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