Compute the average value of \( f(t) = \sin(t) + \cos(t) \) over the interval \([0, \pi]\).

put \( z=\mathrm{Re}^{i \theta} \), then \( |z|=R \) Now, \( \left|\int_{\Gamma} f(z) d z\right| \leq \frac{\pi R}{\left(R^{2}-1\right)^{2}} \rightarrow 0 \) as \( R \rightarrow \infty \) Thus, the equation (1) becomes \( \lim _{R \rightarrow \infty} \int_{-R}^{R} f(x) d x+\lim _{R \rightarrow \infty} \int_{\Gamma} f(z) d z=\frac{\pi}{2} \) \( \Rightarrow \quad \int_{-\infty}^{\infty} f(x) d x+0=\frac{\pi}{2} \Rightarrow \int_{-\infty}^{\infty} f(x) d x=\frac{\pi}{2} \)

\( \int _ { - \infty } ^ { \infty } \frac { \sin x } { x } d x \)

(iv) Find the equation of the tangent to the curne \( y=x^{3}-x \) at \( x=2 \)

\( \int _ { 0 } ^ { \infty } \int _ { 0 } ^ { \infty } \frac { e ^ { - y } } { y } d y d x \)

f) Solve : \( \left(1+y^{2}\right) d x+x d y=\tan ^{-1} y d y \)