Evaluate the following integrals please

**yeeho** · October 14th 2010, 12:35 PM

hello, can someone help me these questions?

a) {{{ int ( lnt/sqrt(t), dt, 4, 9) }}}
b) {{{ int ( cos(sqrt(6x)), dx ) }}}

**pickslides** · October 14th 2010, 01:02 PM

Originally Posted by yeeho

a) {{{ int ( lnt/sqrt(t), dt, 4, 9) }}}

For $\int \frac{\ln t}{\sqrt{t}}~dt$

By parts, make $\displaystyle v = \ln t , \frac{du}{dt} = \frac{1}{\sqrt{t}}$

Originally Posted by yeeho

b) {{{ int ( cos(sqrt(6x)), dx ) }}}

For this one sub $t = \sqrt{x}$ then integrate by parts.

**TheCoffeeMachine** · October 14th 2010, 01:44 PM

One way of stating the formula for integration by parts is:

$\displaystyle \int_a^b f'(x)g(x)\;{dx} = g(x)\cdot f(x)\bigg|_{a}^{b}-\int_a^b f(x)g'(x)\;{dx}$

Therefore (without defining new variables) we can write the integral as:

$\displaystyle \int_4^9 \frac{\ln{t}}{\sqrt{t}}\;{dt} = 2\int_4^9 (\sqrt{t})'\ln{t}}\;{dt} = 2\sqrt{t}\ln{t}\bigg|_4^9-2\int (\ln{t})'\sqrt{t}\;{dt}$

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