Question

integration of tan@​

Answer 1

Answer:

- log cosθ Or log Secθ

Step-by-step explanation:

To find-----> ∫ tanθ dθ

Solution-----> Let ,

I = ∫ tanθ dθ

= ∫ ( Sinθ / Cosθ ) dθ

= ∫ Sinθ dθ / Cosθ

Let, Cosθ = t

=> - Sinθ dθ = dt

=> Sinθ dθ = - dt

I = ∫ - dt / t

= - ∫ dt / t

= - logt + C

Putting t = Cosθ , in it , we get,

= - log Cosθ + C

= - 1 × log Cosθ + C

We know that, n logx = log xⁿ , applying it we get,

= log ( Cosθ )⁻¹ + C

= log ( 1 / Cosθ ) + C

We know that, 1 / Cosθ = Secθ , we get,

= log ( Secθ ) + C

Answer 2

♣ Qᴜᴇꜱᴛɪᴏɴ :

$\bf{\int tan \left(\theta\right)d\theta}}$

♣ ᴀɴꜱᴡᴇʀ :

$\boxed{\bf{\int tan \left(\theta\right)d\theta}\bf{=-\ln \left|cos \left(\theta\right)\right|+C}}$

♣ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

$\mathrm{Use\:the\:following\:identity}:\quad \tan \left(x\right)=\dfrac{\sin \left(x\right)}{\cos \left(x\right)}$

$=\int \dfrac{\sin \left(\theta\right)}{\cos \left(\theta\right)}d\theta$

$\text { Apply u - substitution: } u=\cos (\theta)$

$\sf{=\int \:-\dfrac{1}{u}du}$

$\mathrm{Take\:the\:constant\:out}:\quad \int a\cdot f\left(x\right)dx=a\cdot \int f\left(x\right)dx$

$=-\int \dfrac{1}{u}du$

$\mathrm{Use\:the\:common\:integral}:\quad \int \dfrac{1}{u}du=\ln \left(\left|u\right|\right)$

$=-\ln \left|u\right|$

$\mathrm{Substitute\:back}\:u=\cos \left(\theta\right)$

$=-\ln \left|\cos \left(\theta\right)\right|$

$\mathrm{Add\:a\:constant\:to\:the\:solution}$

$\boxed{\bf{=-\ln \left|\cos \left(\theta\right)\right|+C}}$