\( \int \left( \sin^{-1} x + \cos^{-1} x \right) dx\) = \(\underline{\hspace{2cm}}\)
Show Hint
Solution and Explanation
Step 1: Using trigonometric identity.
We know that \( \sin^{-1} x + \cos^{-1} x = \frac{\pi}{2} \), so the integral becomes:
\[
\int \left( \frac{\pi}{2} \right) dx = \frac{\pi}{2} x + C.
\]
Step 2: Conclusion.
Thus, \( \int \left( \sin^{-1} x + \cos^{-1} x \right) dx = \frac{\pi}{2} x + C \).
Top MPBSE Class XII Board Higher Mathematics Questions
- In the function \( f : \mathbb{R} \rightarrow \mathbb{R} \), given by \( f(x) = 5x \):
- \( \tan^{-1} \sqrt{3} - \sec^{-1}(-2) \) is equal to:
If \( y = \sqrt{e^x} \), \( x > 0 \), then \( \frac{dy}{dx} = \underline{\hspace{2cm}} \)
- Rate of change of area of circle per second with respect to its radius \( r \) when \( r = 5 \, \text{cm} \) will be \(\underline{\hspace{2cm}}\)
- The number of arbitrary constants in the particular solution of a differential equation of third order are \(\underline{\hspace{2cm}}\)
Top MPBSE Class XII Board Integration Questions
- \( \int e^x [f(x) + f'(x)] dx \)= \(\underline{\hspace{2cm}}\)
Match the pairs correctly:
(i) \( \int \tan x \, dx \) \(\hspace{3.5cm}\) \( \log |\sin x| + c \)
(ii) \( \int \cot x \, dx \) \(\hspace{3.5cm}\) \( \log |\csc x| - \cot x + c \)
(iii) \( \int \sec x \, dx \) \(\hspace{3.5cm}\) \( \log |\sec x + \tan x| + c \)
(iv) \( \int \csc x \, dx \) \(\hspace{3.5cm}\) \( -\log |\csc x + \cot x| + c \)
(v) \( \int \frac{\cos x}{\sin x} \, dx \) \(\hspace{3.5cm}\) \( \log |\sin x| + c \)
(vi) Derivative of \( \sin 2x \) with respect to \( x \) \(\hspace{0.75cm}\) \( 2 \cos 2x \)- Evaluate \[ \int_{-1}^{1} \sin^5x \cos^4x \, dx \]
- Evaluate \[ \int_{-1}^{2} \left( x^3 - |x| \right) dx. \]
- Evaluate the integral: \[ \int_0^{\pi} \frac{x \sin x}{1 + \cos^2 x} \, dx \]
Top MPBSE Class XII Board Questions
- What is the Market Economy?
- What is positive economics analysis?
- What is profit?
- What is average product?
- Write three relationships between total utility and marginal utility.