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Solve the trigonometric integral $\int\tan\left(x\right)^3\sec\left(x\right)dx$

Step-by-step Solution

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Solution

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Final Answer

$\frac{-4+12\tan\left(\frac{x}{2}\right)^2}{3\left(1+\tan\left(\frac{x}{2}\right)\right)^{3}\left(1-\tan\left(\frac{x}{2}\right)\right)^{3}}+C_0$
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Step-by-step Solution

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We can solve the integral $\int\tan\left(x\right)^3\sec\left(x\right)dx$ by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of $t$ by setting the substitution

$t=\tan\left(\frac{x}{2}\right)$

Learn how to solve implicit differentiation problems step by step online.

$t=\tan\left(\frac{x}{2}\right)$

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Learn how to solve implicit differentiation problems step by step online. Solve the trigonometric integral int(tan(x)^3sec(x))dx. We can solve the integral \int\tan\left(x\right)^3\sec\left(x\right)dx by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of t by setting the substitution. Hence. Substituting in the original integral we get. Simplifying.

Final Answer

$\frac{-4+12\tan\left(\frac{x}{2}\right)^2}{3\left(1+\tan\left(\frac{x}{2}\right)\right)^{3}\left(1-\tan\left(\frac{x}{2}\right)\right)^{3}}+C_0$

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Solve integral of tanx^3secxdx using basic integralsSolve integral of tanx^3secxdx using u-substitutionSolve integral of tanx^3secxdx using integration by parts

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Function Plot

Plotting: $\frac{-4+12\tan\left(\frac{x}{2}\right)^2}{3\left(1+\tan\left(\frac{x}{2}\right)\right)^{3}\left(1-\tan\left(\frac{x}{2}\right)\right)^{3}}+C_0$

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x
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(◻)
+
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◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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Main Topic: Implicit Differentiation

Implicit differentiation makes use of the chain rule to differentiate implicitly defined functions. For differentiating an implicit function y(x), defined by an equation R(x, y) = 0, it is not generally possible to solve it explicitly for y(x) and then differentiate. Instead, one can differentiate R(x, y) with respect to x and y and then solve a linear equation in dy/dx for getting explicitly the derivative in terms of x and y.

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