MHT CET · Maths · Indefinite Integration
If \(\int \frac{5 \tan x}{\tan x-2} d x=x+a \log |\sin x-2 \cos x|+c\), then \(a\) (Where \(\mathrm{c}\) is constant of integration)
- A 1
- B -2
- C -1
- D 2
Answer & Solution
Correct Answer
(D) 2
Step-by-step Solution
Detailed explanation
Let \(I=\int \frac{5 \tan x}{\tan x-2} d x\)
\(
I=\int \frac{5 \sin x}{\sin x-2 \cos x} d x
\)
Here \(\frac{d}{d x}(\sin x-2 \cos x)=\cos x+2 \sin x\)
\(\therefore \mathrm{I} =\int \frac{(2 \sin x+2 \sin x+\sin x)+(2 \cos x-2 \cos x)}{\sin x-2 \cos x} d x \)
\( =\int \frac{(2 \sin x+\cos x)+(2 \sin x+\cos x)+(\sin x-2 \cos x)}{\sin x-2 \cos x} d x \)
\( =\int \frac{2(2 \sin x+\cos x)+(\sin x-2 \cos x)}{\sin x-2 \cos x} d x \)
\( =\int d x+2 \int \frac{2 \sin x+\cos x}{\sin x-2 \cos x} d x \)
\( =x+2 \log |\sin x-2 \cos x|+c\)
From given data, \(a=2\)
\(
I=\int \frac{5 \sin x}{\sin x-2 \cos x} d x
\)
Here \(\frac{d}{d x}(\sin x-2 \cos x)=\cos x+2 \sin x\)
\(\therefore \mathrm{I} =\int \frac{(2 \sin x+2 \sin x+\sin x)+(2 \cos x-2 \cos x)}{\sin x-2 \cos x} d x \)
\( =\int \frac{(2 \sin x+\cos x)+(2 \sin x+\cos x)+(\sin x-2 \cos x)}{\sin x-2 \cos x} d x \)
\( =\int \frac{2(2 \sin x+\cos x)+(\sin x-2 \cos x)}{\sin x-2 \cos x} d x \)
\( =\int d x+2 \int \frac{2 \sin x+\cos x}{\sin x-2 \cos x} d x \)
\( =x+2 \log |\sin x-2 \cos x|+c\)
From given data, \(a=2\)
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
- The equation of a circle which has a tangent \(3 x+4 y=6\) and two normals given by \((x-1)(y-2)=0\) isMHT CET 2011 Medium
- If \(x=\log t, y+1=\frac{1}{t}, \quad\) then \(e^{-x} \frac{d^{2} x}{d y^{2}}+\frac{d x}{d y}=\)MHT CET 2020 Easy
- \(\int \frac{\sin x+\sin ^3 x}{\cos 2 x} \mathrm{~d} x=\mathrm{A} \cos x+\mathrm{B} \log \mathrm{f}(x)+\mathrm{c}\)
(where \(\mathrm{c}\) is a constant of integration). Then values of A, B and \(\mathrm{f}(x)\) areMHT CET 2023 Hard - If \(\sqrt{\frac{x}{y}}+\sqrt{\frac{y}{x}}=4\), then \(\frac{d y}{d x}=\)MHT CET 2020 Medium
- The arithmetic mean of marks in Mathematics for four divisions A, B, C and D were 80, 75, 70 and 72 respectively. Their standard deviations were \(12,6,8\) and and 10 respectively. Then, division ‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾ has more uniformity.MHT CET 2021 Easy
- A particle moves along a curve \(y=\frac{2 x^3-1}{3}\). The points on the curve at which the \(y\) co-ordinate is changing 18 times the \(x\) co-ordinate areMHT CET 2025 Medium
More PYQs from MHT CET
- Two coils P and Q are kept near each other. When no current flows through coil P and current increases in coil Q at the rate \(10 \mathrm{~A} / \mathrm{s}\), the emf in coil P is 12 mV . When coil Q carries no current and current of 1.5 A flows through coil P , the magnetic flux linked with the coil Q in mWb isMHT CET 2025 Medium
- Given below are two statements:
Statement I - Reactions involved in Krebs cycle are anabolic and catabolic.
Statement II - During oxidation of acetyl Co-A stepwise oxidation of acetyl part of acetyl Co-A occurs.
In light of above statements, choose the most appropriate answer from the option given below.MHT CET 2024 Easy - The mass of the lift is 200 kg , when it ascends with an acceleration of \(4 \mathrm{~m} / \mathrm{s}^2\) then the tension in the cable supporting the lift will be [Given: Acceleration due to gravity \(\left.g=10 \mathrm{~m} / \mathrm{s}^2\right]\)MHT CET 2024 Easy
- The common principal solution of the equations \(\sin \theta=-\frac{1}{2}\) and \(\tan \theta=\frac{1}{\sqrt{3}}\) isMHT CET 2025 Easy
- Identify the chiral molecule from following.MHT CET 2023 Easy
- Which of the following phenols is isolated from defensive secretion of grasshopper species?MHT CET 2020 Easy