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  • please solve rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 1 maths textbook solution

please solve rd sharma class 12 chapter Indefinite Integrals exercise 18.2 question 1 maths textbook solution

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\frac{6}{5} x^{\frac{5}{2}}+\frac{1}{2} x^{\frac{3}{2}}+5 x+c

Hint: To solve this, we break square root then imply \int x^{a} d x formula 

\begin{aligned} &\text { Given: } \int(3 x \sqrt{x}+4 \sqrt{x}+5) d x \\ &\text { Solution: } 3 x x^{\frac{1}{2}}+4 x^{\frac{1}{2}}+5 \\ &=3 x^{\frac{3}{2}}+4 x^{\frac{1}{2}}+5 \\ &I=\int(3 x \sqrt{x}+4 \sqrt{x}+5) d x=\int\left(3 x . x^{\frac{1}{2}}+4 x^{\frac{1}{2}}+5\right) d x \\ &\left\{\int x^{a} d x=\frac{1}{a+1} x^{a+1}+c, a \neq-1\right\} \end{aligned}

\begin{aligned} &=3 \int x^{\frac{3}{2}} d x+4 \int x^{\frac{1}{2}} d x+\int 5 d x \\ &=3 \frac{1}{1+\frac{3}{2}} x^{1+\frac{3}{2}}+4 \frac{1}{1+\frac{1}{2}} x^{1+\frac{1}{2}}+5 x+c \\ &=3 \frac{1}{\frac{5}{2}} x^{\frac{5}{2}}+4 \frac{1}{\frac{3}{2}} x^{\frac{3}{2}}+5 x+c \\ &=\frac{6}{5} x^{\frac{5}{2}}+\frac{8}{3} x^{\frac{3}{2}}+5 x+c \end{aligned}

 

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