The antiderivative of tanx is loge|secx|+C where C is a constant. In this post, let us learn how to find the antiderivative of tanx.
The (integral or) antiderivative formula of tanx is given by
∫tanx dx = ln|secx| + C,
where ln denotes the logarithm with base e and C is an integral constant.
Find the Antiderivative of tanx
The antiderivative of tanx is a function of x whose derivative is tanx. Thus, the antiderivative of tanx is the integration of tanx, that is,
∫ tanx dx.
To find it, let us put
ln secx = t
Differentiating both sides with respect to x, we have
$\dfrac{1}{\sec x}$ secx tanx = $\dfrac{dt}{dx}$
⇒ tanx dx = dt
So we have:
∫tanx dx
= ∫ dt
= t + C
= ln secx + C
As ln secx is defined for positive values of secx, we deduce that
∫tanx dx = ln|secx|+C.
So the antiderivative of tanx is ln|secx|+C where C is an integral constant.
Verification:
Lets verify that the derivative of ln|secx|+C is the function tanx. The derivative of ln|secx|+C is equal to
d/dx(ln|secx|+C)
= tanx + 0
= tanx, hence verified.
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FAQs
Q1: What is the antiderivative of tanx?
Answer: The antiderivative of tan x is ln|secx|+C where C is a constant of integration.