KCET · Maths · Indefinite Integration
If '\(n\)' is a natural number, then \(\int \dfrac{\sin^n x}{\cos^{n+2} x}\,dx = \)
- A \(\dfrac{\tan^{n-1} x}{n - 1} + C\)
- B \(\dfrac{\tan^n x}{n} + C\)
- C \(\dfrac{\tan^{n+2} x}{n + 2} + C\)
- D \(\dfrac{\tan^{n+1} x}{n + 1} + C\)
Answer & Solution
Correct Answer
(D) \(\dfrac{\tan^{n+1} x}{n + 1} + C\)
Step-by-step Solution
Detailed explanation
The given integral is \(I = \int \dfrac{\sin^n x}{\cos^{n+2} x} \,dx\)
This can be rewritten as:
\(I = \int \left(\dfrac{\sin x}{\cos x}\right)^n \cdot \dfrac{1}{\cos^2 x} \,dx\)
\(I = \int \tan^n x \sec^2 x \,dx\)
Let \(\tan x = t\), then \(\sec^2 x \,dx = dt\)
Substituting these into the integral, we get:
\(I = \int t^n \,dt\)
\(I = \dfrac{t^{n+1}}{n+1} + C\)
\(I = \dfrac{\tan^{n+1} x}{n+1} + C\)
Answer: \(\dfrac{\tan^{n+1} x}{n + 1} + C\)
This can be rewritten as:
\(I = \int \left(\dfrac{\sin x}{\cos x}\right)^n \cdot \dfrac{1}{\cos^2 x} \,dx\)
\(I = \int \tan^n x \sec^2 x \,dx\)
Let \(\tan x = t\), then \(\sec^2 x \,dx = dt\)
Substituting these into the integral, we get:
\(I = \int t^n \,dt\)
\(I = \dfrac{t^{n+1}}{n+1} + C\)
\(I = \dfrac{\tan^{n+1} x}{n+1} + C\)
Answer: \(\dfrac{\tan^{n+1} x}{n + 1} + C\)
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
- A simple graph contains 24 edges. Degree of each vertex is 3 . The number of vertices isKCET 2010 Easy
- When \(x>0\), then \(\int \cos ^{1}\left(\frac{1-x^{2}}{1+x^{2}}\right) d x\) isKCET 2011 Medium
- If \(\left|\begin{array}{lll}\mathrm{x}+1 & \mathrm{x}+2 & \mathrm{x}+\mathrm{a} \\ \mathrm{x}+2 & \mathrm{x}+3 & \mathrm{x}+\mathrm{b} \\ \mathrm{x}+3 & \mathrm{x}+4 & \mathrm{x}+\mathrm{c}\end{array}\right|=0\), then \(a, b, c\) areKCET 2009 Medium
- The coordinates of the point on the \(\sqrt{x}+\sqrt{y}=6\) at which the tangent is equally inclined to the axes isKCET 2022 Hard
- Equation of the plane perpendicular to the line \( \frac{x}{1}=\frac{y}{2}=\frac{z}{3} \) and passing through the point \( (2, \),
\( 3,4) \) isKCET 2014 Easy - The domain of the function \(f(x)=\frac{1}{\log _{10}(1-x)}+\sqrt{x+2}\) isKCET 2022 Easy
More PYQs from KCET
- The angle between the line \(x+y=3\) and the line joining the points \((1,1)\) and \((-3,4)\) isKCET 2024 Easy
- Which cleansing agent gets precipitated in hard water?KCET 2019 Hard
- The general solution of \(\sin x-\cos x=\sqrt{2}\), for any integer \(n\) isKCET 2013 Medium
- Oxidative decarboxylation occurs during the formation ofKCET 2013 Medium
- A resistor has a colour code of green, blue, brown and silver. What is its resistance?KCET 2011 Easy
- If \(f(x)=1+n x+\frac{n(n-1)}{2} x^2+\frac{n(n-1)(n-2)}{6}\) \(x^3+\ldots+x^n\), then \(f^n(1)\) is equal toKCET 2023 Medium