Question:
If \( \int x^2 \cos^3 x\, dx = \frac{1}{6}f(x) + g(x) \sin 2x + h(x) \cos 2x + c \), then \( f(1) + g(2) + h\left(\frac{1}{2}\right) = \)
If \( \int x^2 \cos^3 x\, dx = \frac{1}{6}f(x) + g(x) \sin 2x + h(x) \cos 2x + c \), then \( f(1) + g(2) + h\left(\frac{1}{2}\right) = \)
Show Hint
Substitution of values in provided integral forms can yield the result without full integration.
Updated On: Jun 4, 2025
- 0
- 2
- 1
- -1
Hide Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Given form of integration already simplifies the structure, so directly plug into the function:
Use values:
\[
f(1) = 1,\quad g(2) = 0,\quad h(1/2) = 1 \Rightarrow \text{Sum} = 1 + 0 + 1 = 2
\]
Was this answer helpful?
0
0
Top Questions on Integration
- Find: \[ \int \frac{2x}{(x^2 + 3)(x^2 - 5)} \, dx \]
- If \( \int \frac{1}{2x^2} \, dx = k \cdot 2x + C \), then \( k \) is equal to:
- Evaluate \( \int_0^{\frac{\pi}{2}} \frac{x}{\cos x + \sin x} \, dx \)
- Evaluate : \[ I = \int_0^{\frac{\pi}{4}} \frac{dx}{\cos^3 x \sqrt{2 \sin 2x}} \]
- Find: \[ \int \frac{\sqrt{x}}{1 + \sqrt{x^{3/2}}} \, dx \]
View More Questions
Questions Asked in AP EAPCET exam
- If a steel rod of a radius 10 mm and length 80 cm is streched by a force of 66 kN along its length, then the longitudinal stress on the rod is nearly
- AP EAPCET - 2025
- mechanical properties of solids
- The number of all five-letter words (with or without meaning) having at least one repeated letter that can be formed by using the letters of the word INCONVENIENCE is:
- AP EAPCET - 2025
- Binomial Expansion
- If \(\alpha, \beta, \gamma\) are the roots of the equation \[ x^3 - 13x^2 + kx + 189 = 0 \] such that \(\beta - \gamma = 2\), then find the ratio \(\beta + \gamma : k + \alpha\).
- In a container of volume 16.62 m$^3$ at 0°C temperature, 2 moles of oxygen, 5 moles of nitrogen and 3 moles of hydrogen are present, then the pressure in the container is (Universal gas constant = 8.31 J/mol K)
- AP EAPCET - 2025
- Ideal gas equation
- If \[ A = \begin{bmatrix} x & 2 & 1 \\ -2 & y & 0 \\ 2 & 0 & -1 \end{bmatrix}, \] where \( x \) and \( y \) are non-zero real numbers, trace of \( A = 0 \), and determinant of \( A = -6 \), then the minor of the element 1 of \( A \) is:}
- AP EAPCET - 2025
- Complex numbers
View More Questions