How do you complete the proof for the integration of dx/sqrt(a^2 - x^2) = arcsin (x/a) + C.
a is a constant.
Substitute x = a sint(t) in
dx/sqrt(a^2 - x^2)
dx = a cos(t) dt
sqrt(a^2 - x^2) = a |cos(t)|
Because x is between -a and a, t can be chose to be between -pi/2 and pi/2, which means that cos(t) is postive, so we can omit the absolute value signs. This means that:
dx/sqrt(a^2 - x^2) = dt
The integral is thus t + C=
arcsin (x/a) + C
To complete the proof for the integration of dx/sqrt(a^2 - x^2) = arcsin (x/a) + C, we can use a trigonometric substitution.
Step 1: Start by letting x = a*sin(t), where -pi/2 ≤ t ≤ pi/2.
Step 2: Find dx by taking the derivative of both sides of x = a*sin(t) with respect to t. The derivative of sin(t) is cos(t), so we have dx = a*cos(t)*dt.
Step 3: Substitute x = a*sin(t) and dx = a*cos(t)*dt into the original integral:
∫dx/sqrt(a^2 - x^2) = ∫(a*cos(t)*dt)/sqrt(a^2 - a^2*sin^2(t)).
Step 4: Simplify the expression:
∫dx/sqrt(a^2 - x^2) = ∫(a*cos(t)*dt)/sqrt(a^2*(1 - sin^2(t))) = ∫(a*cos(t)*dt)/sqrt(a^2*cos^2(t)).
Step 5: Cancel the a^2 term:
∫dx/sqrt(a^2 - x^2) = ∫(cos(t)*dt)/|cos(t)|.
Step 6: Simplify by removing the absolute value:
∫dx/sqrt(a^2 - x^2) = ∫dt = t.
Step 7: Find t in terms of x by using the trigonometric substitution:
x = a*sin(t) ⟹ t = arcsin(x/a).
Step 8: Finally, the solution is:
∫dx/sqrt(a^2 - x^2) = arcsin(x/a) + C,
where C is the constant of integration.
To prove the integration of dx/sqrt(a^2 - x^2) = arcsin (x/a) + C, we can follow these steps:
Step 1: Start with the given integral: ∫dx/√(a^2 - x^2)
Step 2: To simplify the integral, we can make use of a trigonometric substitution. Let x = a sin(θ), where θ is a new variable.
Step 3: Now, differentiate both sides of x = a sin(θ) with respect to θ to find dx in terms of dθ: dx = a cos(θ) dθ.
Step 4: Substitute the expressions for dx and x into the integral from Step 1:
∫dx/√(a^2 - x^2) = ∫(a cos(θ) dθ)/√(a^2 - a^2 sin^2(θ)).
Step 5: Simplify the expression further:
∫dx/√(a^2 - x^2) = a∫cos(θ) dθ/√(a^2(1 - sin^2(θ)))
= a∫cos(θ) dθ/√(a^2 cos^2(θ))
= a∫(cos(θ) dθ)/|a cos(θ)|.
Step 6: Cancel out the common terms of |a cos(θ)| in the integral:
∫dx/√(a^2 - x^2) = ∫dθ/|cos(θ)|.
Step 7: Recognize that 1/|cos(θ)| is equal to sec(θ), the reciprocal of the cosine function.
Step 8: Rewrite the integral as:
∫dθ/|cos(θ)| = ∫sec(θ) dθ.
Step 9: The integral of sec(θ) can be found using the logarithmic identity for secant:
∫sec(θ) dθ = ln|sec(θ) + tan(θ)| + C.
Step 10: Replace θ with arcsin(x/a) in the result from Step 9 to obtain the final expression:
∫dx/√(a^2 - x^2) = ln|sec(arcsin(x/a)) + tan(arcsin(x/a))| + C.
Step 11: Simplify the expression using the trigonometric identity of secant and tangent in terms of arcsin:
sec(arcsin(u)) = 1/cos(arcsin(u)) = 1/√(1 - u^2),
tan(arcsin(u)) = sin(arcsin(u))/cos(arcsin(u)) = u/√(1 - u^2).
Step 12: Substitute the values of sec(arcsin(x/a)) and tan(arcsin(x/a)) into the integral from Step 10:
∫dx/√(a^2 - x^2) = ln|1/√(1 - (x/a)^2) + x/(a√(1 - (x/a)^2))| + C.
Step 13: Simplify further:
∫dx/√(a^2 - x^2) = ln|1/√(1 - (x^2/a^2)) + x/√(a^2 - x^2)| + C.
Step 14: Finally, use the trigonometric identity √(1 - sin^2(θ)) = cos(θ) to simplify the expression:
∫dx/√(a^2 - x^2) = ln|1/√(1 - (x^2/a^2)) + x/√(a^2 - x^2)| + C
= ln|1/√(1 - (x^2/a^2)) + √(1 - (x^2/a^2)) x/a| + C.
And that completes the proof of the integration of dx/√(a^2 - x^2) = arcsin(x/a) + C.