Immediate indefinite integrals

Text

Compute the following indefinite integrals ($a$ constant):

• (a) $\displaystyle\int\dfrac{4}{x^{4}}\,dx$
• (b) $\displaystyle\int\left(x^{2}+a\right)dx$
• (c) $\displaystyle\int 6x^{-5/2}\,dx$
• (d) $\displaystyle\int\left(5x^{3}+3x^{2}+2x+1\right)dx$
• (e) $\displaystyle\int\left(\sqrt{x}+\sqrt[3]{x}\right)2a^{2}\,dx$

Solution

Use $\displaystyle\int x^{n}\,dx=\dfrac{x^{n+1}}{n+1}+C$ (with $n\neq-1$).

• (a) $-\dfrac{4}{3x^{3}}+C$
• (b) $\dfrac{x^{3}}{3}+a x+C$
• (c) $-4x^{-3/2}+C=-\dfrac{4}{\sqrt{x^{3}}}+C$
• (d) $\dfrac{5}{4}x^{4}+x^{3}+x^{2}+x+C$
• (e) $a^{2}\left(\dfrac{4}{3}x^{3/2}+\dfrac{3}{2}x^{4/3}\right)+C$ $\boxed{-\tfrac{4}{3x^{3}};\ \tfrac{x^{3}}{3}+ax;\ -\tfrac{4}{\sqrt{x^{3}}};\ \tfrac54x^{4}+x^{3}+x^{2}+x;\ a^{2}\!\left(\tfrac43x^{3/2}+\tfrac32x^{4/3}\right)}$