What is the antiderivative of #sec^5xtan^7x#?

Question

Elijah Abram · Accepted Answer

The answer is: #sec^11x/11-sec^9x/3+(3sec^7x)/7-sec^5x/5+c#.

Remembering that:

#=sin^2x/cos^2x+cos^2x/cos^2x=tan^2x+1rArr#

#tan^2x=sec^2x-1#;

#intsecxtanxdx=secx+c#;

#int[f(x)]^n*f'(x)dx=[f(x)]^(n+1)/(n+1)+c#.

Than:

#intsec^5x*tan^7xdx=intsec^5x*tan^6x*tanxdx=#

#=intsec^5x*(sec^2x-1)^3*tanxdx=#

#=intsec^5x(sec^6x-3sec^4x+3sec^2x-1)*tanxdx=#

#=int(sec^11xtanx-3sec^9xtanx+3sec^7xtanx-sec^5xtanx)dx=#

#=intsec^10xsecxtanxdx-3intsec^8xsecxtanxdx+#

#+3intsec^6xsecxtanxdx-intsec^4xsecxtanxdx=#

#=sec^11x/11-(3sec^9x)/9+(3sec^7x)/7-sec^5x/5+c=#

#=sec^11x/11-sec^9x/3+(3sec^7x)/7-sec^5x/5+c#.

Isaiah Allen · Answer

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(The antiderivative of sec^5(x)tan^7(x) is (1/The antiderivative of sec^5(x)tan^7(x) is:

(1/The antiderivative of sec^5(x)tan^7(x) is (1/3The antiderivative of sec^5(x)tan^7(x) is:

(1/6The antiderivative of sec^5(x)tan^7(x) is (1/3)The antiderivative of sec^5(x)tan^7(x) is:

(1/6)The antiderivative of sec^5(x)tan^7(x) is (1/3)secThe antiderivative of sec^5(x)tan^7(x) is:

(1/6)secThe antiderivative of sec^5(x)tan^7(x) is (1/3)sec^The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(xThe antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(xThe antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tanThe antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tanThe antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(xThe antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(xThe antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x)The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x)The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) +The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) +The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)secThe antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)secThe antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(xThe antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)sec(xThe antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(x)The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)sec(x)The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(x)tanThe antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)sec(x)tanThe antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(x)tan^The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)sec(x)tan^The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(x)tan^8The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)sec(x)tan^8The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(x)tan^8(xThe antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)sec(x)tan^8(xThe antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(x)tan^8(x)The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)sec(x)tan^8(x)The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(x)tan^8(x) +The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)sec(x)tan^8(x) +The antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(x)tan^8(x) + CThe antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)sec(x)tan^8(x) + CThe antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(x)tan^8(x) + C,The antiderivative of sec^5(x)tan^7(x) is:

(1/6)sec^3(x)tan^6(x) + (1/30)sec(x)tan^8(x) + CThe antiderivative of sec^5(x)tan^7(x) is (1/3)sec^3(x)tan^6(x) + (1/6)sec(x)tan^8(x) + C, where C is the constant of integration.