MHT CET · Maths · Definite Integration
\(\int_{0}^{\pi / 2} \frac{\sin x-\cos x}{1-\sin x \cdot \cos x} d x\) is equal to
- A 0
- B \(\frac{\pi}{2}\)
- C \(\frac{\pi}{4}\)
- D \(\pi\)
Answer & Solution
Correct Answer
(A) 0
Step-by-step Solution
Detailed explanation
Let \(I=\int_{0}^{\pi / 2} \frac{\sin x-\cos x}{1-\sin x \cos x} d x\)
On putting \(x=\left(\frac{\pi}{2}-x\right)\) in Eq. (i), we get
\(
\begin{aligned}
I &=\int_{0}^{\pi / 2} \frac{\sin \left(\frac{\pi}{2}-x\right)-\cos \left(\frac{\pi}{2}-x\right)}{1-\sin \left(\frac{\pi}{2}-x\right) \cos \left(\frac{\pi}{2}-x\right)} d x \\
&=\int_{0}^{\pi / 2} \frac{\cos x-\sin x}{1-\sin x \cos x} d x \\
&=-\int_{0}^{\pi / 2}\left(\frac{\sin x-\cos x}{1-\sin x \cos x}\right) d x
\end{aligned}
\)
On adding Eqs. (i) and (ii), we get
\(
2 I=\int_{0}^{\pi / 2} 0 d x=0
\)
\(\Rightarrow \quad I=0\)
On putting \(x=\left(\frac{\pi}{2}-x\right)\) in Eq. (i), we get
\(
\begin{aligned}
I &=\int_{0}^{\pi / 2} \frac{\sin \left(\frac{\pi}{2}-x\right)-\cos \left(\frac{\pi}{2}-x\right)}{1-\sin \left(\frac{\pi}{2}-x\right) \cos \left(\frac{\pi}{2}-x\right)} d x \\
&=\int_{0}^{\pi / 2} \frac{\cos x-\sin x}{1-\sin x \cos x} d x \\
&=-\int_{0}^{\pi / 2}\left(\frac{\sin x-\cos x}{1-\sin x \cos x}\right) d x
\end{aligned}
\)
On adding Eqs. (i) and (ii), we get
\(
2 I=\int_{0}^{\pi / 2} 0 d x=0
\)
\(\Rightarrow \quad I=0\)
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
- \(\int \frac{d x}{32-2 x^2}=A \log (4-x)+B \log (4+x)+c\), then the value of \(\mathrm{A}\) and \(\mathrm{B}\) are respectively (where \(\mathrm{c}\) is a constant of integration)MHT CET 2021 Medium
- The 1. Order and Degree of the differential equation \(\sqrt{\frac{\mathrm{dy}}{\mathrm{d} x}}-4 \frac{\mathrm{dy}}{\mathrm{d} x}-7 x=0\) is respectivelyMHT CET 2025 Easy
- \(\int \frac{\log \left(x^2+\mathrm{a}^2\right)}{x^2} \mathrm{~d} x=\)MHT CET 2023 Medium
- The area of the rectangle having vertices \(\mathrm{P}, \mathrm{Q}, \mathrm{R}, \mathrm{S}\) with position vectors \(-\hat{i}+\hat{j}+\hat{k}, \hat{i}+\hat{j}+\hat{k}, \hat{i}-\hat{j}+\hat{k},-\hat{i}-\hat{j}+\hat{k}\) respectively isMHT CET 2025 Easy
- If planes and include angle then the value of isMHT CET 2018 Hard
- A random variable X has the following probability distribution :
\(\begin{array}{|c|c|c|c|c|} \hline \mathrm{X}=x & 1 & 2 & 3 & 4 \\ \hline \mathrm{P}(\mathrm{X}=x) & 0 \cdot 1 & 0 \cdot 2 & 0 \cdot 3 & 0 \cdot 4 \\ \hline \end{array}\)
The mean and standard deviation of X are respectivelyMHT CET 2025 Medium
More PYQs from MHT CET
- In the given circuit, when \(S_1\) is closed, the capacitor \(C\) gets full charged. Then \(S_1\) is kept open and \(S_2\) is closed. Hence
MHT CET 2022 Hard - If L. P. P. has optimum solutions at two consecutive corner points of feasible region, then L. P. P. hasMHT CET 2020 Medium
- Which of the following amines can not be prepared by Gabriel phthalimide synthesis?MHT CET 2020 Medium
- Which among the group -15 elements does NOT exist as tetra atomic molecule?MHT CET 2018 Easy
- Identify the correct increasing order of field strength of ligands form following.MHT CET 2025 Easy
- Action of salivary amylase stops when it mixes with gastric juice in stomach because ___________.MHT CET 2024 Hard