The given integral is:
∫‌‌dx We can factor the denominator as a difference of squares:
x4−16=(x2−4)(x2+4)=(x−2)(x+2)(x2+4) Now, we can use partial fraction decomposition to rewrite the integrand:
To find the constants A, B, C, and D, we multiply both sides by the common denominator:
x=A(x+2)(x2+4)+B(x−2)(x2+4)+(Cx+D)(x−2)(x+2)
We can solve for the constants by plugging in specific values of
x and comparing coefficients.
For example, setting
x=2, we get:
2=A(4)(8)⇒A=‌Similarly, setting
x=−2, we get:
−2=B(−4)(8)⇒B=‌ Setting
x=0, we get:
0=8A−8B−4D⇒D=0Finally, comparing the coefficient of
x∧3 on both sides, we get:
0=A+B+C⇒C=−‌Therefore, the integral becomes:
Now we can integrate each term separately:
‌∫‌‌dx=‌‌ln|x−2|+C1‌∫‌‌dx=‌‌ln|x+2|+C2‌∫‌‌dx=−‌‌ln(x2+4)+C3Combining all the terms, we get:
Using the properties of logarithms, we can simplify the result:
Simplifying further:
∫‌‌dx=‌‌ln‌|‌|+CTherefore, the correct answer is Option C:
‌‌log‌|‌|+C