KCET · Maths · Definite Integration
\(\int\limits_{a-6}^{b-6} f(x + 6)\,dx\) is equal to
- A \(\int\limits_{a}^{b} f(x + 6)\,dx\)
- B \(\int\limits_{a}^{b} f(x - 6)\,dx\)
- C \(\int\limits_{a}^{b} f(x)\,dx\)
- D \(-\int\limits_{a}^{b} f(x)\,dx\)
Answer & Solution
Correct Answer
(C) \(\int\limits_{a}^{b} f(x)\,dx\)
Step-by-step Solution
Detailed explanation
Let \(t = x + 6\).
Differentiating both sides with respect to \(x\), we get \(dt = dx\).
When \(x = a - 6\), \(t = a - 6 + 6 = a\).
When \(x = b - 6\), \(t = b - 6 + 6 = b\).
Substituting these into the integral, we get:
\(\int_{a-6}^{b-6} f(x + 6) \, dx = \int_{a}^{b} f(t) \, dt\)
Using the property of definite integrals \(\int_{a}^{b} f(t) \, dt = \int_{a}^{b} f(x) \, dx\), we get:
\(\int_{a}^{b} f(x) \, dx\)
Answer: \(\int\limits_{a}^{b} f(x)\,dx\)
Differentiating both sides with respect to \(x\), we get \(dt = dx\).
When \(x = a - 6\), \(t = a - 6 + 6 = a\).
When \(x = b - 6\), \(t = b - 6 + 6 = b\).
Substituting these into the integral, we get:
\(\int_{a-6}^{b-6} f(x + 6) \, dx = \int_{a}^{b} f(t) \, dt\)
Using the property of definite integrals \(\int_{a}^{b} f(t) \, dt = \int_{a}^{b} f(x) \, dx\), we get:
\(\int_{a}^{b} f(x) \, dx\)
Answer: \(\int\limits_{a}^{b} f(x)\,dx\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- If \(\overrightarrow{\mathbf{a}}+2 \overrightarrow{\mathbf{b}}+3 \overrightarrow{\mathbf{c}}=\overrightarrow{\mathbf{0}}\), then \(\overrightarrow{\mathbf{a}} \times \overrightarrow{\mathbf{b}}+\overrightarrow{\mathbf{b}} \times \overrightarrow{\mathbf{c}}+\overrightarrow{\mathbf{c}} \times \overrightarrow{\mathbf{a}}\) is equal toKCET 2009 Hard
- If \(3 x^{2}+x y-y^{2}-3 x+6 y+k=0\) represents \(a\) pair of lines, then \(k\) is equal toKCET 2008 Hard
- The order and degree of the differential equation \(y=\frac{d p}{d x} x=\sqrt{a^{2} p^{2}+b^{2}}, \quad\) where \(\mathrm{p}=\frac{\mathrm{dy}}{\mathrm{dx}}\) (here \(\mathrm{a}\) and \(\mathrm{b}\) are arbitrary constants) respectively areKCET 2010 Easy
- Which of the following statements is not correct?KCET 2025 Easy
- In the figure
Statement-I: When \(\alpha > \beta \geq 0\), the section is hyperbola
Statement-II: When \(\beta > 90^\circ\), the section is ellipse
Which of the following is correct?KCET 2026 Medium - The area of the region bounded by the curve \(y^{2}=8 x\) and the line \(y=2 x\) isKCET 2020 Medium
More PYQs from KCET
- Sertoli cells are nourshing cells in the testis. They also secrete a hormone. Identify the same.KCET 2005 Hard
- Function of potassium ethyl xanthate in froth floatation process is to make the oreKCET 2020 Easy
- Consider the following statements regarding the network shown in the figure.
(1) The equivalent resistance of the network between points \(A\) and \(B\) is independent of value of \(G\).
(2) The equivalent resistance of the network between points \(A\) and \(B\) is \(\frac{4}{3} R\)
(3) The current through \(G\) is zero.
Which of the above statements is/are true?KCET 2011 Medium - The trajectory of a projectile projected from origin is given by the equation \( y=x-\frac{2 x^{2}}{5} \). The initial velocity of the projectile isKCET 2019 Hard
- \( \int_{0}^{\frac{\pi}{2}} \frac{1}{a^{2} \cdot \sin ^{2} x+b^{2} \cdot \cos ^{2} x} d x \) is equal toKCET 2017 Medium
- A spring is stretched by applying a load to its free end. The strain produced in the spring isKCET 2016 Easy