Calculus
∫ ( x 7 + x 2 + 1 ) 3 7 x 1 3 + 5 x 1 5 d x = a 1 ⋅ ( x 7 + x 2 + 1 ) c x b
Given that the indefinite integral above is true, what's the value of a + b + c , where a , b , and c are positive integers?
The answer is 18.
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3 solutions
Hi Pranav! Very nice solution ! What I did was differentiate the Right hand side and matched the respective powers .
I did the same thing Jit did... :P Very nice solution indeed!
hmm.............. that is correct answer
Can you tell me how did you upload such Equations??
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Its by using LaTeX.. https://brilliant.org/math-formatting-guide/
could anyone tell me how to post links??
correct
take the variable with highest power outside in these kind of problems
Very elegant solution!
did it the same way :P
In sums like these, the rule is to divide by the greatest power of x, like here it is x^21. Then, you're bound to get the numerator as the derivative of the denominator if a human has given you this question!
TAKE X^7 COMMON FROM DENOMINATOR, THEN PUT 1+X^-5+X^-7= T. IN NUMINATOR WE VE ITS DERIVATIVE.
These Problem becomes interesting when x^7 is taken out of the Denominator. Then we are left with x^21 in Denominator and multiplying it is function consisting of [1+1/(x^5)+1/(x^7)]^3. Taking x^21 in Numerator we get [7/(x^8)+5/(x^6)]. Noticing that the Derivative of [1+1/(x^5)+1/(x^7)] is present in numerator with a Minus sign so lets take it as P. so Integration becomes =-dp/p^3. So answer comes out to be 1/2p^2 +C (where C is constant of Integration) Solving it further and replacing the value of p we are left with
(1/2)*[(x^14)/(x^7+x^2+1)^2]
So we get A=2 B=14 C=2 So A+B+C= 2+14+2=18
put x =1/t
Rewrite the given integral as:
∫ ( 1 + x 5 1 + x 7 1 ) 3 x 8 7 + x 6 5 d x
Use the substitution:
1 + x 5 1 + x 7 1 = t ⇒ ( x 8 7 + x 6 5 ) d x = − d t
Solve the resulting integral and substitute t to obtain
2 1 ( x 7 + x 2 + 1 ) 2 x 1 4
Hence, a = 2 , b = 1 4 , c = 2 ⇒ a + b + c = 1 8 .