What is the integral of t^2+tsqrt(t-1)+tsin(pi)t ?

The integral of t^2+tsqrt(t-1)+tsin(pi)t is (t^3)/3+2/5 (t-1)^{5/2}+2/3 (t-1)^{3/2}+C

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integral of t^2+tsqrt(t-1)+tsin(pi)t

\int t^{2} + t t - 1 + t sin (π) t d t

\frac{t ^{3}}{3} + \frac{2}{5} (t - 1)^{\frac{5}{2}} + \frac{2}{3} (t - 1)^{\frac{3}{2}} + C

Solution steps

\int t^{2} + t t - 1 + t sin (π) t d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int t^{2} d t + \int t t - 1 d t + \int t sin (π) t d t

\int t^{2} d t = \frac{t ^{3}}{3}

\int t t - 1 d t = \frac{2}{5} (t - 1)^{\frac{5}{2}} + \frac{2}{3} (t - 1)^{\frac{3}{2}}

\int t sin (π) t d t = 0

= \frac{t ^{3}}{3} + \frac{2}{5} (t - 1)^{\frac{5}{2}} + \frac{2}{3} (t - 1)^{\frac{3}{2}} + 0

Simplify

= \frac{t ^{3}}{3} + \frac{2}{5} (t - 1)^{\frac{5}{2}} + \frac{2}{3} (t - 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{t ^{3}}{3} + \frac{2}{5} (t - 1)^{\frac{5}{2}} + \frac{2}{3} (t - 1)^{\frac{3}{2}} + C

\int_{- 1}^{2} (x^{2} - 4 x + 3) d x

\frac{d}{d x} (\frac{1}{2} x + 3)

\int e^{2 x} - \frac{1}{2} e^{\frac{x}{2}} d x

\frac{d}{d x} (e^{- x} + \frac{x}{3})

x \to \infty lim (∣ x ∣)

What is the integral of t^2+tsqrt(t-1)+tsin(pi)t ?
The integral of t^2+tsqrt(t-1)+tsin(pi)t is (t^3)/3+2/5 (t-1)^{5/2}+2/3 (t-1)^{3/2}+C