Find value of following integral ∫xtan⁡(x)sec⁡(x)(tan⁡(x)−x)2dxthe numerator is d[sec⁡(x)] but that isnt work due to x in denominator. First we can simplify as∫xsin⁡(x)(sin⁡(x)−xcos⁡(x))2dx=∫x(sin⁡(x)−xcos⁡(x))dx+∫x2cos⁡(x)(sin⁡(x)−xcos⁡(x))2dxbut again its not manipulative. Suggest a useful substitution or method.

Question

titanspokey20urvn · Accepted Answer

Hint:For ∫xsin⁡(x)dx(sin⁡(x)−xcos⁡(x))2d(sin⁡x−xcos⁡x)dx=cos⁡x−(cos⁡x−xsin⁡x)=?

izvozna39g0 · Accepted Answer

Consider the function f(x)=1(sin⁡x−xcos⁡x) . Once differentiation gives:f′(x)=−(cos⁡x+xsin⁡x−cos⁡x)(sin⁡x−xcos⁡x)2=−xsin⁡x(sin⁡x−xcos⁡x)2So your integral is:∫xsin⁡x(sin⁡x−xcos⁡x)2dx=−1(sin⁡x−xcos⁡x)

Find value of following integral \(\displaystyle\int{\frac{{{x}{\tan{{\left({x}\right)}}}{\sec{{\left({x}\right)}}}}}{{{\left({\tan{{\left({x}\right)}}}-{x}\right)}^{{2}}}}}{\left.{d}{x}\right.}\) the numerator

Answered question

Answer & Explanation

New Questions in Integral Calculus