Question

integrate cos theta×(tan theta + sec theta) ​

Answer 1

$\displaystyle \sf{ \int \: cos \: \theta(tan\: \theta + sec\: \theta)\: d\theta } = - cos \: \theta + \theta + c$

Given :

$\displaystyle \sf{ \int \: cos \: \theta(tan\: \theta + sec\: \theta)\: d\theta }$

To find :

Integrate the integral

Solution :

Step 1 of 2 :

Write down the given Integral

Here the given Integral is

$\displaystyle \sf{ \int \: cos \: \theta(tan\: \theta + sec\: \theta)\: d\theta }$

Step 2 of 2 :

Integrate the integral

$\displaystyle \sf{ \int \: cos \: \theta(tan\: \theta + sec\: \theta)\: d\theta }$

$\displaystyle \sf{ = \int cos \: \theta \bigg( \frac{sin \: \theta}{cos \: \theta} + \frac{1}{cos \: \theta} \bigg )\: d\theta }$

$\displaystyle \sf{ = \int \bigg( \frac{sin \: \theta \times cos \: \theta}{cos \: \theta} + \frac{cos \: \theta}{cos \: \theta} \bigg )\: d\theta }$

$\displaystyle \sf{ = \int ( sin \: \theta + 1)\: d\theta }$

$\displaystyle \sf{ = \int sin \: \theta \: d\theta +\int 1\: d\theta }$

$\displaystyle \sf{ = - cos \: \theta \: d\theta + \theta + c }$

Where c is integration constant

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Answer 2

Answer:

∫ cosθ(tanθ+secθ) dθ= θ-cosθ + +c

Explanation:

Given:

  cos theta×(tan theta + sec theta) ​

to find:

   integrate cos theta×(tan theta + sec theta) ​

solution:

   ∫ cosθ(tanθ+secθ) dθ

     =∫(cosθ(sinθ/cosθ +1/cosθ) dθ

multiplying cosθ inside the bracket,we get

     = ∫ (sinθ +1)dθ

     =∫ (sinθdθ +dθ)

      =∫ (sinθdθ) + ∫dθ

      = -cosθ + θ+c

where c is the constant of integration.

∫ cosθ(tanθ+secθ) dθ= θ-cosθ + +c

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