Evaluate: `int[f(x)g^(x)-f^(x)g(x)]dx`

Evaluate int[f(x)g^(n)(x)-f^(n)(x)g(x)]dx

Evaluate: inte^x(f(x)+f^(prime)(x))dx=e^xf(x)+C

Evaluate int_(0)^(a)(f(x))/(f(x)+f(a-x)) dx.

Let f and g be continuous fuctions on [0, a] such that f(x)=f(a-x)" and "g(x)+g(a-x)=4 " then " int_(0)^(a)f(x)g(x)dx is equal to

Evaluate: ifintg(x)dx=g(x),t h e nintg(x){f(x)+f^(prime)(x)}dx

If f: RrarrR and g:RrarrR are two given functions, then prove that 2min.{f(x)-g(x),0}=f(x)-g(x)-|g(x)-f(x)|

By using the properties of definite integrals, evaluate the integrals Show that int_(0)^(a)f(x)g(x)dx=2int_(0)^(a)f(x)dx , if f and g are defined as f(x)=f(a-x) and g(x)+g(a-x)=4 .

If ("lim")_(xveca)[f(x)+g(x)]=2a n d ("lim")_(xveca)[f(x)-g(x)]=1, then find the value of ("lim")_(xveca)f(x)g(x)dot

Using the first principle, prove that: d/(dx)(f(x)g(x))=f(x)d/(dx)(g(x))+g(x)d/(dx)(f(x))