Integral Calculus
Definite Integrals
MMTS_Full_Test_02
Grade 12
Question:
The value of the definite integral $\displaystyle\int_{-2}^{2}x^3\ln(1^x+3^x+5^x+15^x)\,dx$
$\dfrac{\ln 15}{4}$
$\dfrac{64}{5}\ln 15$
$\dfrac{32}{5}\ln 15$
$\dfrac{64}{5}\ln 30$
Step-by-Step Solution
Key Concept: Split ln term; use odd/even function property
$\ln(1^x+3^x+5^x+15^x)=\ln[(1+3^x)(1+5^x)]$. Use King's property and symmetry. Answer $=\frac{32}{5}\ln 15$.
Correct Answer: 3