Trigonometric integrals. Evaluate the following integrals. ∫sin3xcos5xdx

Question

Trigonometric integrals. Evaluate the following integrals.  ∫sin3xcos5xdx

Wendy Boykin · Accepted Answer

Step 1 Given: The integral ∫sin3xcos5xdx To evaluate: the integration of given trigonometric integral. Step 2 Explanation: Let I=∫sin3xcos5xdx This can be re-written as I=∫sin⁡xsin2xcos5xdx ⇒I=∫sin⁡x(1−cos2x)cos5xdx   [∵sin2x+cos2x=1] ⇒I=∫sin⁡x(cos5x−cos7x)dx Now substituting cos⁡x=t ⇒−sin⁡xdx=dt ⇒sin⁡xdx=−dt Hence I=∫(t5−t7)(−dt) ⇒I=−∫(t5−t7)dt ⇒I=−(t66−t88)+C ⇒I=t88−t66+C Now as t=cos⁡x Therefore I=cos8x8−cos6x6+C where C is arbitrary constant and also known as &quot;constant of integration&quot; Answer: ∫sin3xcos5xdx=cos8x8−cos6x6+C where C is constant of integration.

Bob Huerta · Accepted Answer

∫cos5(x)sin3(x)dx  =∫−cos5(x)(cos2(x)−1)⋅sin⁡(x)dx  =∫u5(u2−1)du  =∫(u7−u5)du  =∫u7du−∫u5du  ∫u7du  =u88  ∫u5du  =u66  ∫u7du−∫u5du  =u88−u66  =cos8(x)8−cos6(x)6  ∫cos5(x)sin3(x)dx  =cos8(x)8−cos6(x)6+C

karton · Accepted Answer

∫sin⁡(x)3cos⁡(x)5dx∫t3−2t5+t7dt∫t3dt−∫2t5dt+∫t7dtt44−t63+t88sin⁡(x)44−sin⁡(x)63+sin⁡(x)88sin⁡(x)44−sin⁡(x)63+sin⁡(x)88+C

Trigonometric integrals. Evaluate the following integrals. \int \sin^{3}x\cos^{5}xdx

Answered question

Answer & Explanation

New Questions in Calculus 2