E N C H A N T E D

I N T E G R A L S

Where Math Meets Magic!
1. ∫01 x³ e⁻²ˣ dx 🔮This spell requires integration by parts! Wave your wand three times.
Step 1: We'll use the magic of integration by parts: ∫u dv = uv - ∫v du
Step 2: Let u = x³ ⇒ du = 3x² dx
Let dv = e⁻²ˣ dx ⇒ v = -½ e⁻²ˣ
Step 3: Apply the magical formula:
∫x³ e⁻²ˣ dx = -½ x³ e⁻²ˣ - ∫(-½ e⁻²ˣ)(3x²) dx
= -½ x³ e⁻²ˣ + ³⁄₂ ∫x² e⁻²ˣ dx
Step 4: Repeat the incantation on ∫x² e⁻²ˣ dx
After two more magical applications, we reveal the complete antiderivative
Step 5: Evaluate from 0 to 1:
= [-½ x³ e⁻²ˣ - ¾ x² e⁻²ˣ - ¾ x e⁻²ˣ - ⅜ e⁻²ˣ] from 0 to 1
Final Answer: = 3/8 - (13/8)e⁻² ≈ 0.0302
2. ∫01 [sin(3 tan⁻¹x) tan⁻¹x] / (1 + x²) dx 🧪This potion requires a substitution spell! Try u = tan⁻¹x
Step 1: Let u = tan⁻¹x ⇒ du = 1/(1 + x²) dx
Step 2: When x = 0, u = 0; when x = 1, u = π/4
Step 3: The potion transforms into:
∫ sin(3u) · u du from 0 to π/4
Step 4: Use the magical identity sin(3u) = 3sin(u) - 4sin³(u)
Step 5: Separate into two potions:
= 3∫u sin(u) du - 4∫u sin³(u) du
Step 6: Stir each potion using integration by parts
Final Answer: = (3√2 π)/16 - (9√2)/16 + 3/4 ≈ 0.123
3. ∫0π/2 [esin⁻¹x sin⁻¹x] / √(1 - x²) dx 📜This ancient scroll needs a substitution spell! Try u = sin⁻¹x
Step 1: Let u = sin⁻¹x ⇒ du = 1/√(1 - x²) dx
Step 2: When x = 0, u = 0; when x = 1, u = π/2
Step 3: The scroll reveals: ∫ u eᵘ du from 0 to π/2
Step 4: Cast integration by parts spell:
Let v = u ⇒ dv = du
Let dw = eᵘ du ⇒ w = eᵘ
Step 5: The magic formula reveals:
∫ u eᵘ du = u eᵘ - ∫ eᵘ du = u eᵘ - eᵘ + C = eᵘ(u - 1) + C
Final Answer: = eπ/2(π/2 - 1) + 1 ≈ 4.810
4. ∫0π/2 x² cos(2x) dx 🔍This mystery requires double integration by parts! Investigate carefully.
Step 1: We'll cast integration by parts spell twice
Step 2: First incantation:
Let u = x² ⇒ du = 2x dx
Let dv = cos(2x) dx ⇒ v = ½ sin(2x)
∫x² cos(2x) dx = ½ x² sin(2x) - ∫x sin(2x) dx
Step 3: Second incantation on ∫x sin(2x) dx:
Let u = x ⇒ du = dx
Let dv = sin(2x) dx ⇒ v = -½ cos(2x)
∫x sin(2x) dx = -½ x cos(2x) + ½ ∫cos(2x) dx
Step 4: Combine magical results:
∫x² cos(2x) dx = ½ x² sin(2x) - [-½ x cos(2x) + ¼ sin(2x)] + C
= ½ x² sin(2x) + ½ x cos(2x) - ¼ sin(2x) + C
Step 5: Reveal the truth from 0 to π/2:
At π/2: ½ (π/2)² (0) + ½ (π/2)(-1) - ¼ (0) = -π/4
At 0: 0 + 0 - 0 = 0
Final Answer: = -π/4 ≈ -0.785