How do you integrate sec^6 (3x) dx?
∫ sec^6(3x) dx = rewrite it as: ∫ sec^4(3x) sec²(3x) dx = rewrite sec^4(3x) in terms of tan(3x) as: ∫ [sec²(3x)]² sec²(3x) dx = ∫ [tan²(3x) + 1]² sec²(3x) dx = expand the square: ∫ [tan^4(3x) + 2tan²(3x) + 1] sec²(3x) dx = substitute tan(3x) = u differentiate both sides: d[tan(3x)] = du 3sec²(3x) dx = du sec²(3x) dx = (1/3)du yielding: ∫ [tan^4(3x) + 2tan²(3x) + 1]² sec²(3x) dx = ∫ (u^4 + 2u² + 1) (1/3) du = break it up and pull constants out: (1/3) ∫ u^4 du + (2/3) ∫ u² + (1/3) ∫ du = (1/3) [1/(4+1)] u^(4+1) + (2/3) [1/(2+1)] u^(2+1) + (1/3)u + C = (1/3)(1/5)u^5 + (2/3)(1/3)u^3 + (1/3)u + C = (1/15)u^5 + (2/9)u^3 + (1/3)u + C substitute back tan(3x) for u, ending with: ∫ sec^6(3x) dx = (1/15) tan^5(3x) + (2/9) tan^3(3x) + (1/3) tan(3x) + C I hope it helps..
What is the integral of cos^5(3x)dx?
∫ cos⁵(3x) dx = being cosine odd-powered, rewrite the integrand as: ∫ cos⁴(3x) cos(3x) dx = being cos²(3x) = 1 - sin²(3x), ∫ [1 - sin²(3x)]² cos(3x) dx = that can be expanded as: ∫ [1 - 2sin²(3x) + sin⁴(3x)] cos(3x) dx = ∫ [cos(3x) - 2sin²(3x)cos(3x)+ sin⁴(3x)cos(3x)] dx = breaking it up, ∫ cos(3x) dx - 2 ∫ sin²(3x)cos(3x) dx + ∫ sin⁴(3x)cos(3x) dx = now let sin(3x) = u → differentiating both sides: d[sin(3x)] = du → 3 cos(3x) = du → cos(3x) = (1/3) du thus, substituting, ∫ cos(3x) dx - 2 ∫ sin²(3x)cos(3x) dx + ∫ sin⁴(3x)cos(3x) dx = ∫ (1/3) du - 2 ∫ u²(1/3) du + ∫ u⁴(1/3) du = (1/3) ∫ du - (2/3) ∫ u² du + (1/3) ∫ u⁴ du = (1/3) u - (2/3) (u²⁺¹)/(2+1) + (1/3) (u⁴⁺¹)/(4+1) + c = (1/3) u - (2/3)(1/3) u³ + (1/3)(1/5) u⁵ + c = (1/3)u - (2/9)u³ + (1/15)u⁵ + c = finally, substituting back u = sin(3x), you get: ∫ cos⁵(3x) dx = (1/3) sin(3x) - (2/9) sin³(3x) + (1/15) sin⁵(3x) + c I hope it has been helpful.. Bye!
Integral of: 3cos^5(3x)dx?
3(cos(3x))^5 = 3 cos(3x) (cos(3x))^4 = 3 cos(3x) (cos(3x)^2)^2 = 3 cos(3x) (1 - sin(3x)^2)^2 use u = sin(3x) du = 3 cos(3x) dx ∫3 cos(3x) (1 - sin(3x)^2)^2 dx = ∫(1 - u^2)^2 du = ∫(1 - 2u^2 + u^4) du u - (2/3) u^3 + (1/5) u^5 = sin(3x) - (2/3) sin^3(3x) + (1/5) sin^5(3x) + c