Which of the following statements are TRUE?
A. \( \int_4^9\displaystyle \frac{1}{\sqrt{x}}\,dx=2 \)
B. \( \int_{-4}^{4}\displaystyle (x^{101}+x^{201}+x^{301})\,dx=4\left(\displaystyle\frac{1}{102}+\displaystyle\frac{1}{202}+\displaystyle\frac{1}{302}\right) \)
C. \( \int_{-1}^{1}\displaystyle x^{1012}\,dx=\displaystyle\frac{2}{1013} \)
D. \( \int\displaystyle (x^2+2x+1)\,dx=\displaystyle\frac{(x+1)^3}{3}+C \) where C is an arbitrary constant
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → A, C and D only
Working out each integral:
\( \displaystyle \int_4^9 x^{-1/2}\,dx=\Big[2\sqrt{x}\Big]_4^9 \)
\( \displaystyle =2(3-2)=2 \)
Each term of \( x^{101}+x^{201}+x^{301} \) is an odd power of x, so the integrand is an odd function. Over the symmetric interval [−4, 4]:
\( \displaystyle \int_{-4}^{4}(x^{101}+x^{201}+x^{301})\,dx=0 \)
\( \displaystyle \int_{-1}^{1}x^{1012}\,dx=\left[\frac{x^{1013}}{1013}\right]_{-1}^{1} \)
\( \displaystyle =\frac{1}{1013}-\left(-\frac{1}{1013}\right)=\frac{2}{1013} \)
\( \displaystyle \int(x^2+2x+1)\,dx=\int(x+1)^2\,dx=\frac{(x+1)^3}{3}+C \)
So the integrals in A, B, C and D are 2, 0, \( \frac{2}{1013} \) and \( \frac{(x+1)^3}{3}+C \).
Checking the statements
(A) \( \int_4^9\displaystyle \frac{1}{\sqrt{x}}\,dx=2 \) — Correct. \( 2\sqrt{9}-2\sqrt{4}=2 \).
(B) \( \int_{-4}^{4}\displaystyle (x^{101}+x^{201}+x^{301})\,dx=4\left(\displaystyle\frac{1}{102}+\displaystyle\frac{1}{202}+\displaystyle\frac{1}{302}\right) \) — Incorrect. The integral of an odd function over [−4, 4] is 0.
(C) \( \int_{-1}^{1}\displaystyle x^{1012}\,dx=\displaystyle\frac{2}{1013} \) — Correct. \( \frac{1}{1013}+\frac{1}{1013}=\frac{2}{1013} \).
(D) \( \int\displaystyle (x^2+2x+1)\,dx=\displaystyle\frac{(x+1)^3}{3}+C \) where C is an arbitrary constant — Correct. The integrand is \( (x+1)^2 \).