integrate cos^10xdx..
.plz show working i really wanna learn these thanks anyway
This has to be done in several steps using integration by parts. To see how it works, take a look at
https://en.wikipedia.org/wiki/Integration_by_reduction_formulae
or, you can expand cos^10(x) as combinations of single powers of cosines:
512 cos^10(x) = 126 + 210cos(2x) + 120cos(4x) + 45cos(6x) + 10cos(8x) + cos(10x)
those terms are easy to integrate. They come from repeated use of
cos(2x) = 2 cos^2(x)-1
cos(4x) = 2cos^2(2x)-1
and so on.
To integrate cos^10(x)dx, you can use the power-reduction identities and the binomial expansion.
Step 1: Apply the power-reduction formula for cosine:
cos^2(x) = (1/2)(1 + cos(2x))
Step 2: Apply the power-reduction formula again:
cos^4(x) = (1/2)^2(1 + cos(2x))^2 = (1/4)(1 + 2cos(2x) + cos^2(2x))
Step 3: Substitute the expression for cos^2(x):
cos^4(x) = (1/4)(1 + 2cos(2x) + (1/2)(1 + cos(4x)))
Step 4: Apply the power-reduction formula one more time:
cos^8(x) = [(1/4)(1 + 2cos(2x) + (1/2)(1 + cos(4x)))]^2
Step 5: Simplify the expression:
cos^8(x) = (1/16)(1 + 4cos(2x) + 4cos^2(2x) + 2cos(4x) + cos^2(4x))
Step 6: Substitute the expression for cos^2(x) once more:
cos^8(x) = (1/16)(1 + 4cos(2x) + 4[(1/2)(1 + cos(4x))] + 2cos(4x) + cos^2(4x))
Step 7: Expand and simplify the expression:
cos^8(x) = (1/16)(1 + 4cos(2x) + 2 + 2cos(4x) + (1/2) + (1/2)cos(4x) + cos^2(4x))
= (1/16)(15/2 + 4cos(2x) + 3cos(4x) + cos^2(4x))
Step 8: Substitute back into the original integral:
∫cos^10(x) dx = ∫[cos^8(x)] * cos^2(x) dx
= (1/16)∫[(15/2 + 4cos(2x) + 3cos(4x) + cos^2(4x))] dx
Step 9: Distribute the integral:
∫cos^10(x) dx = (1/16)∫[(15/2)*1 dx + (4cos(2x))*cos^2(x) dx + (3cos(4x))*cos^2(x) dx + (cos^2(4x))*cos^2(x) dx]
Step 10: Evaluate each integral separately:
∫(15/2)*dx = (15/2)x + C
∫(4cos(2x))*cos^2(x) dx = 2∫cos(2x)*(1 + cos(2x)) dx
= 2∫[cos(2x) + cos^3(2x)] dx
= 2[(1/2)sin(2x) + (1/12)sin^3(2x)] + C
∫(3cos(4x))*cos^2(x) dx = 0 (This integral evaluates to zero due to an odd number of cosine terms)
∫(cos^2(4x))*cos^2(x) dx = ∫[1/2(1 + cos(8x))]*cos^2(x) dx
= (1/2)∫[cos^2(x) + cos^2(x)*cos(8x)] dx
= (1/2)∫[(1/2)(1 + cos(2x)) + (1/2)(1 + cos(2x))*cos(8x)] dx
= (1/4)∫[(1 + cos(2x) + (1 + cos(2x))*cos(8x)] dx
= (1/4)[(1/2)x + (1/6)sin(2x) + (1/6)sin(2x)*cos(8x)] + C
Step 11: Sum up the individual integrals:
∫cos^10(x) dx = (15/2)x + C + 2[(1/2)sin(2x) + (1/12)sin^3(2x)] + (1/4)[(1/2)x + (1/6)sin(2x) + (1/6)sin(2x)*cos(8x)] + C
Simplifying this expression gives the final result.