Integrating the following expressions with respect to x.
(a) \( \scriptsize 2x^4\)
(b) \( \scriptsize 4x^{\frac{1}{2}}\)
(c) \( \scriptsize 2 \sqrt{x}\)
(d) \( \scriptsize 3x^4 – x + 2\)
(e) \( \frac{ 3x^3 + x – 4}{2x^3}\)
Solution
(a) \( \scriptsize 2x^4\)
\(\scriptsize \int 2x^4 dx = 2 \int x^4 dx \)= \( \frac{2 \left( x^{4 + 1}\right)}{4 + 1} \scriptsize + c \)
= \( \frac{2 x^{5}}{5} \scriptsize + c \)
(b) \( \scriptsize 4x^{\frac{1}{2}}\)
\(\scriptsize \int 4x^{\frac{1}{2}} dx = 4 \int x^{\frac{1}{2}} dx \)= \( \frac{4 \left( x^{\scriptsize\frac{1}{2} \;+ \;1}\right)}{\frac{1}{2} + 1} \scriptsize + c \)
= \( \frac{4 \left(x^{\scriptsize\frac{1}{2} \;+ \;1}\right)}{\frac{3}{2}} \scriptsize + c \)
= \( \frac{4 \; \times \; 2 \left( x^{\scriptsize\frac{1}{2} \;+ \;1}\right)}{3} \scriptsize + c \)
= \( \frac{8 x^{\scriptsize \frac{3}{2}}}{3} \scriptsize + c \)
(c) \( \scriptsize 2 \sqrt{x}\)
\(\scriptsize \int 2 \sqrt{x}dx = 2 \int \sqrt{x} dx = 2 \int \ x^{\frac{1}{2}} dx \)= \( \frac{2 \left( x^{\scriptsize \frac{1}{2} + 1}\right)}{\frac{1}{2} + 1} \scriptsize + c \)
= \( \frac{2 \left( x^{\scriptsize \frac{1}{2} + 1}\right)}{\frac{3}{2}} \scriptsize + c \)
= \( \frac{2 \; \times \; 2( x^{\frac{1}{2} + 1}) }{3} \scriptsize + c \)
= \( \frac{4 x^{\scriptsize \frac{3}{2}}}{3} \scriptsize + c \)
(d) \( \scriptsize 3x^4 – x + 2\)
= \( \scriptsize 3x^4 – x + 2 \\ \scriptsize = \int 3 x^4 dx – \int x dx +\int 2 x dx \)
= \(\scriptsize 3 \int x^4 dx – \int x dx + 2 \int x dx \)
= \( \frac{3 \left( x^{4 + 1}\right)}{4 + 1} – \frac{\left( x^{1 + 1}\right)}{1 + 1} \scriptsize + 2x + c \)
= \( \frac{3 x^5}{5} – \frac{x^2}{2} \scriptsize + 2x + c \)
(e) \( \frac{ 3x^3 + x \;-\; 4}{2x^3}\)
\(\int \normalsize \frac{ 3x^3 + x \;- \;4}{2x^3}\scriptsize dx \\= \normalsize \int \left (\normalsize \frac{3 x^3}{2x^3} + \frac{x}{2x^3} – \frac{4}{2x^3} \right)\scriptsize dx \) \(\int \left (\normalsize \frac{3 }{2} + \frac{x^{-2}}{2} – \scriptsize 2x^{-3} \right)\scriptsize dx \)= \(\frac{3 }{2} \int \scriptsize dx + \normalsize \frac{1 }{2} \int \scriptsize x^{-2} dx \; – \scriptsize 2\int x^{-3} dx \)
= \(\frac{3 }{2} \scriptsize x + \normalsize \frac{1 }{2} \frac{ \left( x^{-2 + 1}\right)}{-2 + 1} \; – \scriptsize 2 \normalsize \frac{ \left( x^{-3 + 1}\right)}{-3 + 1} \scriptsize + c\)
= \(\frac{3 }{2} \scriptsize x \; – \normalsize \frac{1 }{2x} \; + \normalsize \frac{1 }{x^{2}} + c\)
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