🌈 Complete Integral Solver ✨

1 Use reduction formula: \( \int \sin^n x \, dx = -\frac{\sin^{n-1} x \cos x}{n} + \frac{n-1}{n} \int \sin^{n-2} x \, dx \)

2 For definite integrals from 0 to π/2, first term evaluates to 0 at both limits

3 Recursive relation: \( I_n = \frac{n-1}{n} I_{n-2} \), where \( I_n = \int_{0}^{\frac{\pi}{2}} \sin^n x \, dx \)

4 Apply recursively for n=10: \[ I_{10} = \frac{9}{10} I_8 \] \[ I_8 = \frac{7}{8} I_6 \] \[ I_6 = \frac{5}{6} I_4 \] \[ I_4 = \frac{3}{4} I_2 \] \[ I_2 = \frac{1}{2} I_0 \] \[ I_0 = \frac{\pi}{2} \]

5 Multiply all coefficients: \[ \frac{9}{10} \times \frac{7}{8} \times \frac{5}{6} \times \frac{3}{4} \times \frac{1}{2} \times \frac{\pi}{2} \]

6 Simplify step-by-step: \[ \frac{63}{80} \times \frac{5}{6} = \frac{315}{480} = \frac{21}{32} \] \[ \frac{21}{32} \times \frac{3}{4} = \frac{63}{128} \] \[ \frac{63}{128} \times \frac{1}{2} = \frac{63}{256} \]

1 For odd powers, use substitution: Let \( u = \sin x \), \( du = \cos x dx \)

2 Rewrite integral using \( \cos^2 x = 1 - \sin^2 x \): \[ \int (1 - u^2)^3 du \]

3 Expand \( (1 - u^2)^3 \): \[ 1 - 3u^2 + 3u^4 - u^6 \]

4 Integrate term by term from 0 to 1: \[ \int (1 - 3u^2 + 3u^4 - u^6) du \]

5 Calculate antiderivative: \[ u - u^3 + \frac{3}{5}u^5 - \frac{1}{7}u^7 \]

6 Evaluate at limits: \[ \left(1 - 1 + \frac{3}{5} - \frac{1}{7}\right) - 0 = \frac{16}{35} \]

1 Substitute \( u = 2x \), \( du = 2dx \) ⇒ \( dx = \frac{du}{2} \)

2 Change limits: when x=0, u=0; when x=π/4, u=π/2

3 Rewrite integral: \[ \frac{1}{2} \int_{0}^{\frac{\pi}{2}} \sin^6 u \, du \]

4 Use reduction formula: \[ I_6 = \frac{5}{6} I_4 \] \[ I_4 = \frac{3}{4} I_2 \] \[ I_2 = \frac{1}{2} I_0 \] \[ I_0 = \frac{\pi}{2} \]

5 Multiply coefficients: \[ \frac{5}{6} \times \frac{3}{4} \times \frac{1}{2} \times \frac{\pi}{2} = \frac{15}{96}\pi \]

6 Include the 1/2 from substitution: \[ \frac{1}{2} \times \frac{15}{96}\pi = \frac{15}{192}\pi = \frac{5}{64}\pi \]

1 Substitute \( u = 3x \), \( du = 3dx \) ⇒ \( dx = \frac{du}{3} \)

2 Change limits: when x=0, u=0; when x=π/6, u=π/2

3 Rewrite integral: \[ \frac{1}{3} \int_{0}^{\frac{\pi}{2}} \sin^5 u \, du \]

4 Express sin⁵u as sinu(1-cos²u)²: \[ \sin^5 u = \sin u (1 - 2\cos^2 u + \cos^4 u) \]

5 Let \( v = \cos u \), \( dv = -\sin u du \)

6 Transform integral: \[ -\frac{1}{3} \int (1 - 2v^2 + v^4) dv \] from v=1 to v=0

7 Integrate term by term: \[ -\frac{1}{3} \left[ v - \frac{2}{3}v^3 + \frac{1}{5}v^5 \right]_1^0 \]

8 Evaluate: \[ -\frac{1}{3} \left[ 0 - \left(1 - \frac{2}{3} + \frac{1}{5}\right) \right] = \frac{1}{3} \left(1 - \frac{2}{3} + \frac{1}{5}\right) = \frac{8}{45} \]

1 Use identity sin²x = 1 - cos²x: \[ \sin^2 x \cos^4 x = (1 - \cos^2 x)\cos^4 x = \cos^4 x - \cos^6 x \]

2 Split integral: \[ \int (\cos^4 x - \cos^6 x) dx \]

3 Use reduction formula for cosⁿx: \[ I_n = \frac{n-1}{n} I_{n-2} \]

4 Calculate I₄: \[ I_4 = \frac{3}{4} I_2 = \frac{3}{4} \times \frac{1}{2} I_0 = \frac{3\pi}{16} \]

5 Calculate I₆: \[ I_6 = \frac{5}{6} I_4 = \frac{5}{6} \times \frac{3\pi}{16} = \frac{15\pi}{96} = \frac{5\pi}{32} \]

6 Subtract results: \[ \frac{3\pi}{16} - \frac{5\pi}{32} = \frac{\pi}{32} \]

1 Substitute \( u = \frac{x}{4} \), \( du = \frac{dx}{4} \) ⇒ \( dx = 4du \)

2 Change limits: when x=0, u=0; when x=2π, u=π/2

3 Rewrite integral: \[ 4 \int_{0}^{\frac{\pi}{2}} \sin^7 u \, du \]

4 Express sin⁷u as sinu(1-cos²u)³: \[ \sin^7 u = \sin u (1 - 3\cos^2 u + 3\cos^4 u - \cos^6 u) \]

5 Let \( v = \cos u \), \( dv = -\sin u du \)

6 Transform integral: \[ -4 \int (1 - 3v^2 + 3v^4 - v^6) dv \] from v=1 to v=0

7 Integrate term by term: \[ -4 \left[ v - v^3 + \frac{3}{5}v^5 - \frac{1}{7}v^7 \right]_1^0 \]

8 Evaluate: \[ -4 \left[ 0 - \left(1 - 1 + \frac{3}{5} - \frac{1}{7}\right) \right] = 4 \times \frac{16}{35} = \frac{64}{35} \]

1 Rewrite integrand: \[ \sin^3 \theta \cos^5 \theta = \sin \theta (1 - \cos^2 \theta) \cos^5 \theta \]

2 Distribute: \[ \sin \theta \cos^5 \theta - \sin \theta \cos^7 \theta \]

3 Let \( u = \cos \theta \), \( du = -\sin \theta d\theta \)

4 Transform integral: \[ -\int (u^5 - u^7) du \] from u=1 to u=0

5 Integrate term by term: \[ -\left[ \frac{u^6}{6} - \frac{u^8}{8} \right]_1^0 \]

6 Evaluate: \[ -\left[ 0 - \left(\frac{1}{6} - \frac{1}{8}\right) \right] = \frac{1}{6} - \frac{1}{8} = \frac{1}{24} \]

1 Expand (1-x)³: \[ (1-x)^3 = 1 - 3x + 3x^2 - x^3 \]

2 Multiply by x²: \[ x^2 (1 - 3x + 3x^2 - x^3) = x^2 - 3x^3 + 3x^4 - x^5 \]

3 Integrate term by term: \[ \int (x^2 - 3x^3 + 3x^4 - x^5) dx \]

4 Calculate antiderivatives: \[ \frac{x^3}{3} - \frac{3x^4}{4} + \frac{3x^5}{5} - \frac{x^6}{6} \]

5 Evaluate from 0 to 1: \[ \left(\frac{1}{3} - \frac{3}{4} + \frac{3}{5} - \frac{1}{6}\right) - 0 \]

6 Find common denominator (60): \[ \frac{20}{60} - \frac{45}{60} + \frac{36}{60} - \frac{10}{60} \]

7 Combine fractions: \[ \frac{20 - 45 + 36 - 10}{60} = \frac{1}{60} \]