∫ x3(x2-1)1 / 4 d x

Question

∫ x3(x2-1)1 / 4 d x (A)  (2( x 2-1)5 / 4/45)(4+5 x 2)+ C (B) (2( x 2-1)5 / 4/45)(5+4 x 2)+ C (C) ((x2-1)5 / 4/45)(4+x2)+C (D) ((x2-1)4 / 5/45)(4 x2+1)+C

Tardigrade Admin · Accepted Answer

I=∫ x3(x2-1)1 / 4 d x Hint: x 2-1= t 4 ; xdx =2 t 3 dt I =∫ 2(t4+1) t ⋅ t3 d t =2 ⋅ ∫(t8+t4) d t=(2/9) t9+(2 t5/5)=(2/9)(x2-1)9 / 4+(2/5)(x2-1)5 / 4 =(2/45)(x2-1)5 / 4(5 x2+4) ⇒ (A)

Q. $\int x^{3} (x^{2} - 1)^{1/4} d x$

$\frac{2 ( x ^{2} - 1 ) ^{5/4}}{45} (4 + 5 x^{2}) + C$

$\frac{2 ( x ^{2} - 1 ) ^{5/4}}{45} (5 + 4 x^{2}) + C$

$\frac{( x ^{2} - 1 ) ^{5/4}}{45} (4 + x^{2}) + C$

$\frac{( x ^{2} - 1 ) ^{4/5}}{45} (4 x^{2} + 1) + C$

Q. ∫x3(x2−1)1/4dx

452(x2−1)5/4​(4+5x2)+C

452(x2−1)5/4​(5+4x2)+C

45(x2−1)5/4​(4+x2)+C

45(x2−1)4/5​(4x2+1)+C

I=∫x3(x2−1)1/4dx Hint: x2−1=t4;xdx=2t3dt I=∫2(t4+1)t⋅t3dt =2⋅∫(t8+t4)dt=92​t9+52t5​=92​(x2−1)9/4+52​(x2−1)5/4 =452​(x2−1)5/4(5x2+4)⇒(A)

Q. $\int x^{3} (x^{2} - 1)^{1/4} d x$

$\frac{2 ( x ^{2} - 1 ) ^{5/4}}{45} (4 + 5 x^{2}) + C$

$\frac{2 ( x ^{2} - 1 ) ^{5/4}}{45} (5 + 4 x^{2}) + C$

$\frac{( x ^{2} - 1 ) ^{5/4}}{45} (4 + x^{2}) + C$

$\frac{( x ^{2} - 1 ) ^{4/5}}{45} (4 x^{2} + 1) + C$