∫ (x3/x+1) dx is equal to

Question

∫ (x3/x+1) dx is equal to (A) x+ (x2/2) +(x3/3) - log |1-x| +C (B) x+ (x2/2) -(x3/3) - log |1-x| +C (C) x- (x2/2) -(x3/3) - log |1+x| +C (D) x- (x2/2) +(x3/3) - log |1+x| +C

Tardigrade Admin · Accepted Answer

We have, I = ∫ (x3/x+1) dx = ∫( x3 -1 + 1 /x+1) dx = ∫ (((x+1)(x2+1 -x)/x+1) - (1/x+1)) dx = ∫ ( x2 +1 -x) dx -∫ (dx/x+1) = (x3/3) +x - (x2/2)- log |x+1| +C

Q. $\int \frac{x ^{3}}{x + 1} d x$ is equal to

$x + \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - l o g ∣ 1 - x ∣ + C$

$x + \frac{x ^{2}}{2} - \frac{x ^{3}}{3} - l o g ∣ 1 - x ∣ + C$

$x - \frac{x ^{2}}{2} - \frac{x ^{3}}{3} - l o g ∣ 1 + x ∣ + C$

$x - \frac{x ^{2}}{2} + \frac{x ^{3}}{3} - l o g ∣ 1 + x ∣ + C$

We have, $I = \int \frac{x ^{3}}{x + 1} d x = \int \frac{x ^{3} - 1 + 1}{x + 1} d x$

$= \int (\frac{( x + 1 ) ( x ^{2} + 1 - x )}{x + 1} - \frac{1}{x + 1}) d x$

$= \int (x^{2} + 1 - x) d x - \int \frac{d x}{x + 1}$

$= \frac{x ^{3}}{3} + x - \frac{x ^{2}}{2} - l o g ∣ x + 1 ∣ + C$

Q. ∫x+1x3​dx is equal to

x+2x2​+3x3​−log∣1−x∣+C

x+2x2​−3x3​−log∣1−x∣+C

x−2x2​−3x3​−log∣1+x∣+C

x−2x2​+3x3​−log∣1+x∣+C