Antiderivative of tanx For finding the antiderivative of tanx we will use some identities of trigonometry, substitution method and the log identities the antiderivative of tanx is also known as integration of tanx. To get the antiderivative tanx use the trigonometric identity tanx = 1/cotx = sinx/cosx. This implies that tanx = sinx/cosx ( since tanx = sinx/cosx ). It means ∫ tanx dx = ∫ sinx/cosx dx Now use the substitution method of integration for cosx Let cosx = t by differentiating it we get - sinx dx = dt Put the value of sinx and dx in above integral equation we get Such that ∫ tanx dx = ∫ sinx/cosx dx = - ∫1/t dt It implies that ∫ tanx dx = ∫ sinx/cosx dx = - logt +c Where c belongs to a constant By substituting the value of t in the equation we will get ∫ tanx dx = ∫ sinx/cosx dx = - log(cosx) + c Know More About Properties of Rational Numbers
Therefore we get ∫ tanx dx = ∫ sinx/cosx dx = - log(cosx) + c As we know the logarithmic identities like Log m +log n =log (m*n ) Log m – log n = log (m/n) Log mn = n log m So we can say -log (cosx ) can be written as log (1/cosx ) Which can further be written as log (secx) since we know that 1/cosx = secx Therefore log (1/cosx ) = log (secx) So we can say that the antiderivative of tanx can be written as ∫ tanx dx = - log(cosx) + c = log (secx ) +c Where, c is the constant. So, finally we get the result that antiderivative of tanx = -log(cosx) +c where c belongs to a constant. as we seen the antiderivative of tanx become quite easier by the method we used known as substitution method. By other method may be the antiderivative of tanx is quite typical and confusing.
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