Find the indefinite integral. \int \sqrt[3]{\tan x}\sec^{2}xdx

Linda Seales 2021-12-11 Answered

Find the indefinite integral.
tanx3sec2xdx

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Expert Answer

Jenny Sheppard
Answered 2021-12-12 Author has 35 answers

Step 1
We have to solve the indefinite integral:
tanx3sec2xdx
Solving the integral by substitution method.
Assuming,
t=tanx
Differentiating,
dtdx=dtanxdx
=sec2x
dt=sec2xdx
Step 2
Substituting above values in the given integral, we get
tanx3sec2xdx=t3dt
=t13dt
=t13+113+1+C (since xndx=xn+1n+1+C)
=t1+331+33+C
=t4343+C
=34t43+c
=34(tanx)43+C
Hence, value of given indefinite integration is 34(tanx)43+C.

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Elois Puryear
Answered 2021-12-13 Author has 30 answers
Given:
sec2(x)3{tan(x)}dx
Substitution u=tan(x)dudx=sec2(x)
=3{u}du
Integral of a power function:
undu=un+1n+1 at n=13:
=3u434
Reverse replacement u=tan(x):
=3tan43(x)4
sec2(x)3{tan(x)}dx
=3tan43(x)4+C
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