inte^(arcsin x)dx

Question

inte^(arcsin x)dx

Bytelearn · Accepted Answer

Recognize the integral: Recognize the integral to be solved. We need to find the integral of e^(arcsin x) with respect to x, which is written as ∫e^(arcsin x)dx.Use substitution to simplify: Use substitution to simplify the integral. Let u = arcsin x, which implies that x = sin u. Then, we need to find dx in terms of du. Since u = arcsin x, we differentiate both sides with respect to x to get du/dx = 1/√(1 - x^2). Therefore, dx = √(1 - x^2)du.Substitute x and dx: Substitute x and dx in the integral. The integral becomes ∫e^u * √(1 - sin^2 u)du. Since sin^2 u + cos^2 u = 1, we have √(1 - sin^2 u) = cos u. So the integral simplifies to ∫e^u * cos u du.Use integration by parts: Use integration by parts. Integration by parts states that ∫u dv = uv - ∫v du. We let u = e^u and dv = cos u du. Then du = e^u du and v = sin u. Applying integration by parts gives us e^u * sin u - ∫sin u * e^u du.Apply integration by parts again: Apply integration by parts again to the remaining integral. We need to integrate ∫sin u * e^u du. Let's use integration by parts again with u = sin u and dv = e^u du. Then du = cos u du and v = e^u. This gives us sin u * e^u - ∫e^u * cos u du.Notice the repeating integral: Notice the repeating integral. We see that ∫e^u * cos u du appears again. Let's call this integral I. Our equation from Step 4 and Step 5 becomes e^u * sin u - (sin u * e^u - I), which simplifies to e^u * sin u - sin u * e^u + I = I.Solve for I: Solve for I. We have I = e^u * sin u - sin u * e^u + I. Subtracting I from both sides, we get 0 = e^u * sin u - sin u * e^u, which implies that I = e^u * sin u.Substitute back in terms of x: Substitute back in terms of x. We originally let u = arcsin x, so we substitute back to get I = e^(arcsin x) * sin(arcsin x). Since sin(arcsin x) = x, our integral I = e^(arcsin x) * x.Add the constant of integration: Add the constant of integration. The final answer is I = e^(arcsin x) * x + C, where C is the constant of integration.

$\int e^{\arcsin x}\,dx$

Full solution

More problems from Sin, cos, and tan of special angles

Related topics

∫earcsin⁡x dx\int e^{\arcsin x}\,dx∫earcsinxdx

$\int e^{\arcsin x}\,dx$