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Solve the following.
If ∫ x⁻³ 5^(1/(x2)) dx = K 5^(1/(x2)) + C, then find the value of K.
If $\int x^{-3}\, 5^{\frac{1}{x^2}}\,dx = K\, 5^{\frac{1}{x^2}} + C$, then find the value of $K$.
Answer
Answer
Put $t = \dfrac{1}{x^2}$, so $dt = -\dfrac{2}{x^3}dx$ and $x^{-3}dx = -\dfrac{dt}{2}$.
$\int x^{-3}\,5^{1/x^2}dx = -\dfrac{1}{2}\int 5^t\,dt = -\dfrac{1}{2}\cdot\dfrac{5^t}{\ln 5} + C = -\dfrac{1}{2\ln 5}\,5^{1/x^2} + C$.
Hence $K = -\dfrac{1}{2\ln 5}$.
Final answer: $-\dfrac{1}{2\ln 5}$