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Provide Solution For R.D.Sharma Maths Class 12 Chapter 18  Indefinite Integrals Exercise  Revision Exercise Question 76 Maths Textbook Solution.

Answers (1)

Answer:

-\frac{1}{3 \cos ^{8} x}-\cos x-\frac{2}{\cos x}+c

Hint:

To solve the given statement we will split sin 5x into sin? x sin x.

Given:

\int \frac{\sin ^{5} x}{\cos ^{4} x} d x

Solution:

=\int \frac{\sin ^{4} x \sin x}{\cos ^{4} x} d x

   =\int \frac{\left(1-\cos ^{2} x\right)^{2} \sin x}{\cos ^{4} x} d x \quad[\text { put } \cos x=t,-\sin x d x=d t]

=\int \frac{\left(1-t^{2}\right)^{2}(-d t)}{t^{4}}

=\int \frac{\left(1+t^{4}-2 t^{2}\right)(-d t)}{t^{4}}

=\int\left(\frac{1}{t^{4}}+1-\left(\frac{2}{t^{2}}\right)\right)(-d t)

=-\int \frac{1}{t^{4}} d t-\int d t+2 \int \frac{1}{t^{2}} d t

=-\frac{t^{-8}}{-3}-t+2\left(-\frac{t}{t}\right)+c

=-\frac{1}{3 \cos ^{8} x}-\cos x-\frac{2}{\cos x}+c

 

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