Question

Find the following integral : $\int \frac{x^3+3x+4}{\sqrt{x}} \, dx$

Answer 1

Answer:

Step-by-step explanation:

Concept:

$1.\int[f(x)+g(x)]\:dx=\int{f(x)}\;dx+\int{g(x)}\;dx\\\\2.\int{x^n}\:dx=\frac{x^{n+1}}{n+1}+c$

Now,

$\int[\frac{x^3+3x+4}{\sqrt{x}}]\:dx\\\\=\int[\frac{x^3}{\sqrt{x}}+\frac{3x}{\sqrt{x}}+\frac{4}{\sqrt{x}}]\:dx\\\\=\int{x^{\frac{5}{2}}}\:dx+3\int{x^{\frac{1}{2}}}\:dx+4\int{x^{\frac{-1}{2}}}\:dx\\\\=\frac{x^{\frac{7}{2}}}{\frac{7}{2}}+3\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+4\frac{x^{\frac{1}{2}}}{\frac{1}{2}}+c\\\\=\frac{2}{7}x^{\frac{7}{2}}+3.\frac{2}{3}x^{\frac{3}{2}}+4.2.x^{\frac{1}{2}}+c\\\\=\frac{2}{7}x^{\frac{7}{2}}+2x^{\frac{3}{2}}+8x^{\frac{1}{2}}+c$

Answer 2

Topic:

Integration

Solution:

We need to perform the following integration.

$\displaystyle\int\dfrac{x^3 + 3x + 4}{\sqrt{x}}\, dx$

Now we can decompose the fraction into 3 fractions.

$\displaystyle\longrightarrow\int\dfrac{x^3}{\sqrt{x}} + \dfrac{3x}{\sqrt{x}} + \dfrac{4}{\sqrt{x}}\, dx$

$\displaystyle\longrightarrow\int\dfrac{x^2\cdot\sqrt{x}\cdot\sqrt{x}}{\sqrt{x}} + \dfrac{3\sqrt{x}\cdot\sqrt{x}}{\sqrt{x}} + 4 \cdot x^{-1/2}\ dx$

$\displaystyle\longrightarrow\int x^2\sqrt{x} + 3\sqrt{x}+ 4 \cdot x^{-1/2}\ dx$

$\displaystyle\longrightarrow\int x^2\cdot x^{1/2} + 3\cdot x^{1/2}+ 4 \cdot x^{-1/2}\ dx$

$\displaystyle\longrightarrow\int x^{2 + 1/2} + 3\cdot x^{1/2}+ 4 \cdot x^{-1/2}\ dx$

$\displaystyle\longrightarrow\int x^{5/2} + 3\cdot x^{1/2}+ 4 \cdot x^{-1/2}\ dx$

Now here, we can use the following power rule of integration.

$\boxed{\int x^n \, dx = \dfrac{x^{n+1}}{n+1} + C}$

$\displaystyle\longrightarrow\dfrac{x^{5/2 + 1}}{5/2 + 1} + 3\cdot \dfrac{x^{1/2 + 1}}{1/2 + 1}+ 4 \cdot \dfrac{x^{-1/2 + 1}}{-1/2 + 1} + C$

$\displaystyle\longrightarrow\dfrac{x^{7/2}}{7/2} + 3\cdot \dfrac{x^{3/2}}{3/2}+ 4 \cdot \dfrac{x^{1/2}}{1/2} + C$

$\displaystyle\longrightarrow\dfrac{2\sqrt{x^7}}{7} + 2x^{3/2}+ 8x^{1/2}+ C$

$\displaystyle\longrightarrow\dfrac{2x^3\sqrt{x}}{7} + 2x\sqrt{x}+ 8\sqrt{x}+ C$

$\displaystyle\longrightarrow\dfrac{2}{7}\ \sqrt{x}\ (x^3 + 7x+ 28)+ C$

This is the required answer.