# Find the following integrals: a) \int (4x^2 - 1)\,dx b) \int (\sqrt x - \frac 2x)\, dx

## Question:

Find the following integrals:

a) {eq}\int (4x^2 - 1)\,dx {/eq}

b) {eq}\int (\sqrt x - \frac 2x)\, dx {/eq}

## Indefinite Integration:

The integration which is evaluated without the upper and lower limits is known as Indefinite Integration.

The basic identity of Indefinite integration is {eq}\displaystyle\int x^n \ dx=\dfrac{x^{n+1}}{n+1}+C {/eq}, where {eq}C {/eq} is known as the constant of integration.

## Answer and Explanation:

a) {eq}I=\displaystyle\int (4x^2 - 1)\,dx {/eq}

On integrating in terms of {eq}x {/eq}, using the identity {eq}\displaystyle\int x^n \ dx=\dfrac{x^{n+1}}{n+1}+C {/eq}, we get:-

{eq}\boxed{I=\dfrac{4x^3}{3}-x+C} {/eq}

b) {eq}I_1=\displaystyle\int (\sqrt x - \dfrac 2x)\, dx {/eq}

On integrating in terms of {eq}x {/eq}, using the identities {eq}\displaystyle\int x^n \ dx=\dfrac{x^{n+1}}{n+1}+C {/eq} and {eq}\displaystyle\int\dfrac{1}{x} \ dx=\ln|x|+C {/eq}, we get:-

{eq}\boxed{I_1=\dfrac{2x^{\frac{3}{2}}}{3}-2\ln|x|+C} {/eq}

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