Question

Evaluate the following indefinite integrals:

1. ∫ (12x^5 − 6𝑥^3 − 4x + 1/2 ) dx
2. ∫ (√𝑥 + √3)^2 dx
3. ∫ (𝑥^𝑒 − 𝑥/𝑒 + 2𝑒𝑥) dx
4. ∫ (2𝑡+1)(2𝑡−1)/2√𝑡 dx
5. ∫ 8𝑦^3 −1/2𝑦+1 dx

Answer 1

Ques:1:

$\intop$ $(12x^5-6x^3-4x+\dfrac{1}{2})dx$

Distribute the integeration on each term

$\Rightarrow\smallint12x^5dx-\smallint6X^3dx-\smallint4xdx+\smallint\dfrac{1}{2}dx$

$\Rightarrow12\times\dfrac{x^{(5+1)}}{5+1}-6\times\dfrac{x^{(3+1)}}{3+1}-4\times\dfrac{x^{1+1}}{1+1}+\dfrac{1}{2}\times\dfrac{x^{0+1}}{0+1}$

$\Rightarrow$ $12\times\dfrac{x^6}{6}-6\times\dfrac{x^4}{4}-4\times\dfrac{x^2}{2}+\dfrac{1}{2}\times x$

$\Rightarrow2x^6-\dfrac{3}{2}x^4-2x^2+\dfrac{x}{2}$

Ques :2:

$\smallint(\sqrt{x}+\sqrt{3})^2dx$

$\Rightarrow\smallint( x+3+2\sqrt{3}\sqrt{x})dx$

$\Rightarrow \smallint xdx+3\smallint dx+\smallint2\sqrt{3}\times x^\dfrac{1}{2}dx$

$\Rightarrow \dfrac{x^2}{2}+3x+2\sqrt{3}\times\dfrac{x^{\dfrac{1}{2}+1}}{\dfrac{1}{2}+1}$

$\Rightarrow\dfrac{x^2}{2}+3x+2\sqrt{3}\times \dfrac{x^{\dfrac{3}{2}}}{\dfrac{3}{2}}$

Ques::3::

$\smallint( x^e-\dfrac{x}{e}+2\times e \times x)dx$

$\Rightarrow\smallint x^edx-\smallint \dfrac{x}{e}dx+\smallint2exdx$

$\Rightarrow \dfrac{x^{e+1}}{e+1}-\dfrac{x^2}{2e}+ex^2$

Ques::4::

$\smallint\dfrac{(2t+1)(2t-1)}{2\sqrt{t}}dx$

As we have to integerate in terms of $x$ .so the terms containing $t$ can be taken outside the integeration

$\Rightarrow$ $\dfrac{(2t+1)(2t-1)}{2\sqrt{t}}\smallint dx$

$\Rightarrow\dfrac{(4t^2-1)}{2\sqrt{t}}\times x$

Ques::5::

$\smallint(8y^3-\dfrac{1}{2y}+1)dx$

$\Rightarrow8y^3\smallint dx- \dfrac{1}{2y}\smallint dx+\smallint dx$

$\Rightarrow 8y^3\times x-\dfrac{x}{2y}+x$