Get Answers to all your Questions

header-bg qa

Explain solution RD Sharma class 12 Chapter 19 Definite Integrals Exercise Revision Exercise question 14

Answers (1)

Answer: \frac{8}{3}

Given:  \int_{0}^{\pi} \sin ^{3} x(1+2 \operatorname{cox})(1+\cos x)^{2} d x

Hint: Use trigonometric identity and then apply substitution method

Solution:  \int_{0}^{\pi} \sin ^{3} x(1+2 \operatorname{cox})(1+\cos x)^{2} d x

\begin{aligned} &\int_{0}^{\pi} \sin x\left(1-\cos ^{2} x\right)(1+2 \operatorname{cox})(1+\cos x)^{2} d x \\ & \end{aligned}

{\left[\begin{array}{l} \because \sin ^{2} x+\cos ^{2} x=1 \\ \sin ^{2} x=1-\cos ^{2} x \end{array}\right]}



\begin{aligned} &\cos x=t \\ & \end{aligned}

-\sin x d x=d t          (Diff w.r.t to x)

\begin{aligned} &\Rightarrow-\int_{0}^{\pi}\left(1-t^{2}\right)(1+2 t)(1+t)^{2} d x \\ & \end{aligned}

\Rightarrow-\int_{0}^{\pi}\left(1-t^{2}\right)(1+2 t)(1+t)^{2} d t \\

\Rightarrow-\int_{0}^{\pi}\left(1+t^{2}+2 t+2 t+2 t^{3}+4 t^{2}-2 t^{2}-t^{4}-2 t^{3}-2 t^{5}-4 t^{4}\right) d t

\begin{aligned} &\Rightarrow-\int_{0}^{\pi}\left(1+4 t+4 t^{2}-2 t^{3}-5 t^{4}-2 t^{5}\right) d t \\ &\end{aligned}

\Rightarrow-\left(t+2 t^{2}+\frac{4 t^{3}}{3}-\frac{t^{4}}{2}-t^{5}-\frac{t^{6}}{3}\right)_{-1}^{1} \\

\left\{\begin{array}{l} \therefore 0<x<\pi \\ 1<\cos x<-1 \\ 1<t<-1 \end{array}\right\}

\begin{aligned} &\Rightarrow\left(t+2 t^{2}+\frac{4 t^{3}}{3}-\frac{t^{4}}{2}-t^{5}-\frac{t^{6}}{3}\right)_{-1}^{1} \\ & \end{aligned}


\begin{aligned} &\Rightarrow 2-\frac{1}{2}+1-2+\frac{5}{3}+\frac{1}{2} \\ & \end{aligned}


Posted by


View full answer

Crack CUET with india's "Best Teachers"

  • HD Video Lectures
  • Unlimited Mock Tests
  • Faculty Support