Find k value where the function is a pdf

Question

Find k value where the function is a pdf

(a) $kx^6(1-x)^4$, for $0 < x < 1, 0$ otherwise

(b) $kx^2(4-x)^3$, for $0 < x < 4$. $0$ otherwise

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my attempt

(a) $$\int_{0}^{1} kx^6(1-x)^4 dx$$

Do I just solve this on $[0,1]?$

Answer 1

Consider the general case of $$I_{m,n}=\int_0^1 x^m (1-x)^n \,dx$$ Let $$x=\sin^2(t) \implies dx=2\sin(t)\cos(t)\,dt$$ making $$I_{m,n}=2 \int_0^{\frac \pi 2} \sin ^{2 m+1}(t) \cos ^{2 n+1}(t)\,dt$$ and using the usual reduction formula $$I_{m,n}=\frac{\Gamma (m+1)\, \Gamma (n+1)}{\Gamma (m+n+2)}$$