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- WE WANT TO FIND THE INTEGRAL OF E
RAISED TO THE POWER OF 3X x SINE 2X USING AN INTEGRATION FORMULA
OFTEN GIVEN IN THE BACK OF A CALCULUS TEXTBOOK
IN AN INTEGRATION TABLE.
IF WE LOOK HARD ENOUGH
WE SHOULD FIND THESE TWO INTEGRATION FORMULAS HERE,
BUT NOTICE OUR INTEGRAND CONTAINS THE SINE FUNCTION,
THEREFORE WE'LL BE APPLYING THIS FIRST INTEGRAL FORMULA HERE.
LET'S BEGIN BY IDENTIFYING THE KEY VALUES OF "A" AND B.
NOTICE, "A" WOULD BE EQUAL TO THREE
SINCE WE HAVE E RAISED TO THE POWER OF 3X.
AND B IS EQUAL TO TWO SINCE WE HAVE SINE 2X.
SOMETIMES YOU'LL SEE THIS FORMULA WITH X AND DX
INSTEAD OF U AND DU.
THIS IS WRITTEN WITH U AND DU
JUST IN CASE WE HAVE TO PERFORM U SUBSTITUTION
IN ORDER TO INTEGRATE,
BUT NOTICE HERE WE CAN SEE THAT U IS EQUAL TO X.
SO WHEN U IS EQUAL TO X NOTICE THAT DU EQUALS DX,
WHICH TELLS US U SUBSTITUTION IS NOT REQUIRED.
SO NOW WE CAN GO AHEAD AND APPLY THE INTEGRATION FORMULA.
WE'LL HAVE E RAISED TO THE POWER OF AU,
WHICH IS E TO THE 3X DIVIDED BY "A" SQUARED + B SQUARED,
THAT WOULD BE 3 SQUARED + 2 SQUARED
x THE QUANTITY "A" SINE BU,
WHICH WOULD BE 3 SINE 2X - B COSINE BU,
THAT'D BE - 2 COSINE 2X AND THEN + C.
SO WE HAVE E TO THE 3X DIVIDED BY--
THIS WOULD BE 9 + 4 = 13 x 3 SINE 2X - 2 COSINE 2X + C.
THIS WOULD BE OUR ANTI-DERIVATIVE.
I HOPE YOU FOUND THIS HELPFUL.