Evaluate
∫ d x 7 x 2 + 1 x + 7 \displaystyle \int\frac{{\left.{d}{x}\right.}}{{{7}{x}^{{2}}+{1}{x}+{7}}} ∫ 7 x 2 + 1 x + 7 d x .
7 x 2 + 1 x + 7 \displaystyle {7}{x}^{{2}}+{1}{x}+{7} 7 x 2 + 1 x + 7 is an irreducible quadratic since the value of
b 2 − 4 a c \displaystyle {b}^{{2}}-{4}{a}{c} b 2 − 4 a c =
Preview Question 6 Part 1 of 5 is negative.
Transform
7 x 2 + 1 x + 7 \displaystyle {7}{x}^{{2}}+{1}{x}+{7} 7 x 2 + 1 x + 7 to
d ( e ( x + f ) 2 + 1 ) \displaystyle {d}{\left({e}{\left({x}+{f}\right)}^{{2}}+{1}\right)} d ( e ( x + f ) 2 + 1 ) for constants d, e and f by completing the square and doing algebra.
d =
Preview Question 6 Part 2 of 5 e =
Preview Question 6 Part 3 of 5 f =
Preview Question 6 Part 4 of 5
Now make the substitution
u = e ( x + f ) \displaystyle {u}=\sqrt{{{e}}}{\left({x}+{f}\right)} u = e ( x + f ) to obtain an integral of the form
g ∫ d u u 2 + 1 \displaystyle {g}\int\frac{{{d}{u}}}{{{u}^{{2}}+{1}}} g ∫ u 2 + 1 d u for a constant g.
Finally integrate.
∫ d x 7 x 2 + 1 x + 7 \displaystyle \int\frac{{\left.{d}{x}\right.}}{{{7}{x}^{{2}}+{1}{x}+{7}}} ∫ 7 x 2 + 1 x + 7 d x =
Preview Question 6 Part 5 of 5 Submit Try a similar question
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Enter your answer as a number (like 5, -3, 2.2172) or as a calculation (like 5/3, 2^3, 5+4)
Enter your answer as a number (like 5, -3, 2.2172) or as a calculation (like 5/3, 2^3, 5+4)
Enter your answer as a number (like 5, -3, 2.2172) or as a calculation (like 5/3, 2^3, 5+4)
Enter your answer as a number (like 5, -3, 2.2172) or as a calculation (like 5/3, 2^3, 5+4)
Enter your answer as an expression. Example: 3x^2+1, x/5, (a+b)/c